Fuse instead of Relay?
 

I checked my fuse box in the front compartment and it seems that I'm missing the fuel pump relay; instead, there's a fuse (the usual flat ones with 2 heads).

This was probably done by some mechanic that worked on my car during the restoration process, but I have no idea why he did it and the effects it has on engine performance. Is it ok to leave it there or replace it with a proper round relay?

You need to tell us the year and model of car. Assuming you have a CIS car, the earlier models (up to at least 75, maybe a few years more) did not have a relay and the fuel pump started with the turn on the ignition key. Your car may be correctly wired, but we need to know a bit more detail.
(Hint: post your car information in your signature line, that way any time you have a question, everyone will know the basics.)

1989 911

Motronic cars have the fuel pump controlled by the DME under the seat, and the relay lives under the seat with it.

The DME relay gets flaky a lot more often than the round ones - carry a spare.