This is addressed in the film's official novelisation which contains a slightly extended version of Kirk's conversations aboard the Bounty. Mr Scott was making plans to turn the hold into a holding tank, per the Captain's orders, and evidently worked out that he'd need materials from at least the twentieth century (onwards).

"Scott here, admiral. Wi' the proper materials--the proper twentieth-century materials--I'll be able to build ye a tank."

"Thank you, Mr. Scott."

Coupled with the known extinction date of the humpback whale, this gives them a pretty solid date range to aim for with their time warp maneuver. Although they had quite a broad range in mind ("late twentieth century"), Spock's calculations turned out to be reasonably accurate. Given that Perspex® has been readily available from the 1960s, presumably he'd have been just as happy with 1975 as 1999.

"Ready to engage computer, admiral," Spock said.

"What's our target in time?" Jim asked.

"The late twentieth century."

"Surely you can be more specific."

They had a narrow timeframe that would be useful to them.

• They needed to arrive late enough that the whale-tank materials they needed to modify the ship would be readily available -- post-1950s. Not just "possible", but in stock locally.


• They needed an economy robust and entreprenurial enough that their oddball material purchases needs wouldn't pop up any red flags on any government's radar (for instance doing this in the Soviet Union was right out) - so 1950s and up.


• If they had any problems with the vessel's drive systems (and they did), they needed a modern enough society where technology, electronics, nuclear things, etc. were well-developed and readily available (so 1970s at least, and the later the better!)


• 
  Whale science needed to be solidly established, with plenty of experts available - because they needed to catch whales, and they had little chance of getting Klingon sensors to tell them the difference between a gray whale and a sperm whale.
  
  
  • Even better if culture had become affluent enough that there were aquatic zoos like Sea World, so we're definitely into the 1970s.
  
  


• Preferably, racism (including war sentiments) have declined considerably, so Uhura, Sulu, Chekov and Spock will have an easier time, so 1970s and the later the better. It would also help if the Cold War was abating, so 1980s.


• Can't go later than 1991, because in Trek canon, the Eugenics Wars began.

To transport the whales the crew used transparent aluminium for their makeshift holding tank which they had to "invent" themselves and produce using 1985 technology which wouldn't have existed in 900 AD.

I'm not completely familiar with the "time sling-shot" maneuver, but there are multiple possible reasons for that.

Going further back is dangerous

It might be that ~1980 is roughly the earliest year they can safely travel. Traveling any further back might strain the Bird of Prey (also dubbed "rust bucket") too much and you all might die. Any further into the future and you might not have the chance to encounter whales.

Time travel is not an exact science

From what we can see in the movie they never specify to what year they want to go. Quite the opposite, directly after the jump this dialog happens:

Kirk: Picture, please.

[Picture of Earth appears in the viewscreen.]

Kirk: Earth. But when? Spock?

Spock: Judging by the pollution content of the atmosphere, I believe we have arrived at the latter half of the twentieth century.

So they know how to roughly jump back in time, but they can't target a certain year or period.

Time travel is an exact science

In the TOS episode "Tomorrow is yesterday" the crew of the Enterprise learns the mechanics required to jump through time by accident. They complete a maneuver around a black star which hurls them back into the ~1960s on Earth. At the end, they can execute the same maneuver around Sol to travel back to their time.

Unbeknownst to them, this time travel method might have a fixed time span of roughly ~300 years you can jump backwards and forwards with, as the dates neatly line up. Or maybe they knew about it, but it was never established on screen.

Repairs and Extensions

Last but not least, the Bird of Prey is ill equipped to transport whales. That means that you need the resources to remodel it on the fly in the past. That excludes anything before ~1960 I guess, because since then we have made vast advancements in technology. Additionally, Earth is in a pretty bad shape starting from ~1991 with Eugenic War waging and killing tens of million of people, and undoubtedly binding a lot of resources. So you want to land before that, because afterwards you'd either end up in one of three situations:

• You can't acquire the resources required because they are not available because of the war.


• Whales are already extinct.


• The technology has advanced so far that you have a hard time doing your thing.

With these restrictions in mind, the period around ~1980 sounds like quite a good place to be.

Spock, master of spin

Spock put a convincing "spin" on his plan when he explained it to Kirk, playing down the margin of error. Just as Scotty learned to pad his estimates when Kirk was likely to make unreasonable demands, Spock knew that Kirk had trouble relying on logic for making dispassionate decisions in scary situations.

Based on the Star Trek IV transcript, Spock determined the earliest date from which they could return. He listed fuel components as a limiting resource:

KIRK: What is our target in time?

SPOCK: Late twentieth century.

KIRK: Surely you can be more specific.

SPOCK: Not with this equipment. I have had to programme some of the
variables from memory.

KIRK: What are some of the variables?

SPOCK: Availability of fuel components, mass of the vessel through a
time continuum, and the probable location of humpback whales, in this
case, the Pacific basin.

A trip deeper into the past would have presumably required a more extreme slingshot maneuver with greater strain on fuel components.

As it turned out, they arrived right on target, so Spock's computations were accurate. Spock's plan broke the ship, but he had a way to fix it:

SCOTT: Admiral, we have a serious problem. Would you please come down?
It's these Klingon crystals, Admiral. The time-travel drained them.
They're giving out. De-crystallising.

KIRK: Give me a round figure, Mister Scott.

SCOTT: Oh, twenty-four hours, give or take, staying cloaked. After
that, Admiral, we're visible, ...and dead in the water. In any case,
we won't have enough to break out of Earth's gravity, to say nothing
of getting back home.

KIRK: I can't believe we've come this far only to be stopped by this!
Is there no way to re-crystallise dilithium?

SCOTT: Sorry, sir. We can't even do that in the twenty-third century.

SPOCK: Admiral, there may be a twentieth century possibility.

KIRK: Explain.

SPOCK: If memory serves, there was a dubious flirtation with nuclear
fission reactors resulting in toxic side effects. By the beginning of
the fusion era, these reactors had been replaced, but at this time, we
may be able to find some.

KIRK: I thought you said they were toxic.

SPOCK: We could construct a device to collect their high-energy
photons safely. These photons could then be injected into the
dilithium chamber, causing crystalline restructure. ...Theoretically.

KIRK: Where would we find these reactors, ...theoretically?

SPOCK: Nuclear power was widely used in naval vessels.

The way Spock immediately offered a solution to their crystal dilemma (the aforementioned "fuel components", it seems), a suspicious person might think that this was part of his original plan.

(I'm tempted to think that Spock simply remembered reading about the Cetacean Institute's humpback whales, couldn't quite recall all of the details due to post-regeneration trauma, and based the entire plan around a hunch.)