How do I get through this gate in Ustengrav?
You need to use a combination of sprinting and the Whirlwind Sprint shout.
In short: sprint through the three pillars, and as soon as you clear the 3rd pillar (thus opening the 3rd gate), activate your Whirlwind Sprint shout. (And then return to regular sprinting)
This area of the game has scripting to reduce the cooldown of your Whirlwind Sprint so that you're not waiting 20 seconds between attempts, but it is entirely possible to make it through with only a single use of Whirlwind Sprint.
If you're still having trouble (and don't have the Steed Guardian Stone), you might try stripping to your skivvies and giving it another go, as you run marginally faster without armor.