Whether or not there actually is an Anacrusis sprint ability isn't clear because of a very obvious problem as soon as you start playing. Well, not exactly a problem, more just a confusing issue. But as to whether there's a sprint in Anacrusis or not, let us explain what's going on here; why you might be moving slower than you want, and the whole walking around an alien infested ship generally.
Anacrusis sprinting explaining
You'll probably find yourself asking whether there's a sprint in Anacrusis almost as soon as you start playing for one clear reason: the ability to ping objects, items and the best Anacrusis guns in the game lives on the left stick click. That, obviously, is where sprint usually lives which means you'll more than likely ping a map marker several times over as you instinctively click the left stick in, to trigger a non-existent Anacrusis sprint. The sad fact is that there is no sprint, you move at one speed no matter what (for the most part, which we'll get to) and if you need to make some space between you and the enemy, all you've got to use is your bumper shield on L1, which knocks back all enemies in a certain radius.
So that's that: there's no Anacrusis sprint, but you can ping weapons, gear and enemies by pressing the left stick in. Which somehow just makes it more obvious as you litter the level with markers trying to get away from the aliens faster. Just remember, your shield on L1 is your friend if you do need to make some distance between you and the enemies.
However, that's not the end of the Anacrusis movement questions...
Why am I walking so slowly in The Anacrusis?
If you find yourself moving frustratingly slowly there are two reason why you might be creeping along agonizingly in The Anacrusis. The first is simply the alien goo that gets spattered about. Goopers are enemies that fire slime around the level that slows you down if you try and walk through it. You obviously want to avoid that, as getting caught by a ton of angry aliens while you're stuck at half speed is bad. You can at least turn the tables using a Goop Grenade. That fires out a layer of goo that will slow down enemies and not you - a well placed lob in a corridor will give you a perfect slow-mo killzone. And if you're looking through all Anacrusis perks, you'll even see there's some that can help you move through it freely.
The other reason you might be moving slowly in Anacrusis is because you're injured. When your health drops low enough you'll end up limping. Again, this is not ideal, so you either need to stay away from danger or start asking your friends for help.