Tales of Vesperia

When Alexei goes into Overlimit, get next to him and block. He should always use his mystic arte when in Overlimit and afterward he slouches over. The more times he uses his mystic arte, the longer he will stay slouched. It took me a couple times for him to use his mystic arte before I finally got it; But just run up and use all out attacks while Alexei is slouching. Eventually he will go down. Put your allies on defend if his health gets too low and you still haven't done it. Let him heal with his ice move as well if need be.

The key to getting Alexei to use his mystic arte is to attack him until his HP drops to about 40k. Once he reaches this point the best idea is to switch to the Defend strategy so your AI controlled party members won't attack him.. I'd also recommend having Recovery equipped to whichever character you control, as the mystic art is a multi-hit attack that from what I've seen doesn't seem to be blockable and will most likely stagger you.

After Alexei uses his mystic arte he will usually stand still for several seconds. He is NOT vulnerable to being knocked down during this time. Occasionally (for me it was after his fourth mystic arte usage) he will slouch over as if he's tired for a few seconds. When this happens simply run up to him and attack him with a regular attack and he will drop. As with all other secret missions you'll know you've done it correctly because the game will give you a short scene where he falls to the ground and gets back up, then GREAT will appear on the screen.

Something else I found that helps is to switch to a weapon with the Minimum Damage ability once Alexei has taken enough damage to use his mystic arte. That way you can continue to attack him (and refill his OL) without having to worry about killing him.