I've heard someone say that as a sentence fragment interjection a few times, but it's literally the infinitive "to fuck" and then "me". It wouldn't be the most common way of expressing profanity, though. Some of the more likely candidates would be:

Блядь! - (lit. "slut") Damn!
Еб твою мать! - (lit. "I fucked your mother"; sometimes without first word "your mother") FUCK! (really, really so much stronger than English "fuck" in so many ways)
Ебанный в рот! - (lit. "fucked in the mouth") fuck it!
На хуй! - (lit. "onto a cock") to Hell with it, also can mean fuck off

If you wanted to say "fuck me" literally with an imperative form, it would be еби меня. I've heard nearly everything, though. You can hear half of it by watching those videos about the meteorite with sound. It's hilarious how Western news sources didn't think about the fact that they were airing Russian in all its obscene glory. The one guy says, "It sort of fucked across the sky, the cocksucker. What the fuck? The damn windows all fucked out all over the fucking place, into the ass with them." The expressiveness is lost in the translation since the words aren't as uniform in Russian as they sound in English. It's really elegant in a vulgar way.