Rules as Written, Strahd won’t take any damage from moving through sunlight as a legendary action. He also won’t take damage from moving through an area of sunlight during his turn. He (and other vampires) only take damage if they start their turn in sunlight
“Sunlight Hypersensitivity. While in sunlight, Strahd takes 20 radiant damage at the start of his turn“
- Curse of Strahd, emphasis mine
In the context of a round in D&D 5th edition (which is 6 seconds long), this makes sense – if he starts his turn in sunlight, he’s spent at least a couple of seconds exposed to the sun. If he’s just running through, then it’s not long enough to cause damage.
Note that he will still suffer disadvantage on attack rolls and ability checks during this brief exposure.