past/present

Can you use the present tense when describing an incident that occurred in the past?
"He's an incompetent employee. I catch him sleeping in his office last week, and this week, he requests a vacation."

I suppose you can, but perhaps "caught" might me more typical in this case.

This sentence is correct, but it's because of one thing following another, and tends to be used informally as a way of narrating something - as in a play.
It implies a 'I caught him sleeping in his office only last week, and this week he's already asking for a vacation (holiday).' (N.B. 'only' in this case emphasizes the fact that last week was not a long time ago).

More examples:
"I'm in the car, and this lunatic overtakes me doing 100mph. I'm thinking, why's he in such a hurry?"

"He proposes to her on the Monday and before you know it they're married on the Thusday!"