A thoughtful question. No simple answer but we can give you cool information!
Actually, it doesn’t really matter whether the star you are going to is 100 light years away or 1 light year away. These are just incredible, inconceivable distances, and although 99% the speed of light sounds nice, we’re not going to get even a hundredth of 1 percent the speed of light. We just don’t have that technology now or ever, as you’ll see below!
There would be no fuel that could power a spaceship faster and faster toward the goal. At a certain point acceleration would have to stop. You would need to save half of your fuel for slowing down at the other end.
So what you’d have to do is accelerate until half your fuel is burned. Then wait. If that fuel got you half way, then you’d start the retro burn immediately and it would take the same amount of fuel to slow down. If half your fuel took you 1/10 of the way then you’d have to coast for 8/10ths more then the last 1/10 burn the last of your fuel to slow down. To get you going 99% the speed of light would require an almost infinite amount of fuel according to Einstein’s theory of relativity – and then you’d have to slow down too!
It would be a one way trip and you wouldn’t know until you arrived if there was anything out there that could be habitable.
Our closest star, Alpha Centauri, is about 4.4 light years away. That number “4.4” misleads us to think that’s not very far, but it’s further away than you can imagine, and that’s just the closest star. This is a distance of about 5.88 trillion miles away, and there’s another number we can deal with: 5.88 (trillion miles).
The fastest we’ve ever gone was on July 4, 2016, when the Juno spacecraft, assisted by Jupiter’s gravity got up to approximately 165,000 miles per hour (265,000 km/h), breaking all previous space speed records. But even if we could increase that speed 81 times faster than Juno, even if we somehow could come up with the incredible propulsion to both speed up and slow down, that would give us the fantastic speed of 13.3 million mph! Can you even imagine that speed? That happens to be about 2% the speed of light, and it would take us 2,200 years to get to Alpha Centauri without even taking the acceleration or deceleration into account at all. So maybe about 4 or 5 thousand years each way, what to speak of 100 light years like you are asking about would take about 44,000 years each way at 81 times faster than we’ve ever gone before.
Your question was if we could go 99% the speed of light for 100 light years. You did correctly identify that there would be a period of acceleration and an equal period of deceleration at the other end of the trip, but honestly not even a fraction of 1 percent of the speed of light is ever going to be achievable. The weight of the fuel alone would be impossible to move.
We can’t give a number for an answer because we’re never going to get that fast anyway. I did try to put the distances into perspective. We are all living in the world of science fiction and don’t want to accept the numbers. Interstellar travel isn’t in our future.