Welcome to HB, Ben!
I would start by changing the group gaskets. A good source for parts is
Stefano's Espresso Care.
Pavoni Express has well-written instructions on how to do this and just about every other maintenance operation you can imagine. Definitely worth checking out.
Soaking the dispersion screen in a cleaning solution would be a good idea. I'm sure it has quite a bit of residue if the machine has been neglected.
Once you have the group head assembly taken apart, determine if the piston is plastic of brass. If it's plastic, I would upgrade it to brass. Brass pistons hold the gaskets in place better and, in my superstitious opinion, have a better feel to them.
If the owner has been using hard water, you might need to de-scale the boiler. A quick search of the forums should give you some ideas.
As for the olive oil- what have you tried so far? Is it just on the outside, or did it get into the boiler?