IMHO Perfect Dark is one of the oldest shooters that holds up (play the version on the Xbox 360, maybe also the XBone, not sure). Yeah yeah it's a console shooter, but it's outstanding and hi-res + dual-stick controls make it feel modern-enough. Do note that it has
two two-player campaign modes in addition to regular single player: co-op, and...
versus. Yes, versus. Also be aware the difficulty modes change the objectives significantly, and sometimes also your
starting position, so if it seems weirdly short bump up the difficulty.
I think some of the original frantic shooters hold up alright, for what they are. Quake, stuff like that. They're kinda niche genre now, though. Not sure about the early Dooms, hard to judge those as easily. I like them, still, but...
The Half Life remake in the HL2 engine (Black Mesa) is very true to the spirit of the original and smoothes its rough edges off very elegantly, but the original's probably still fairly playable, too.
First mass effect, oof. It suffers from a lot of "which of the three buildings they made will this quest take place in?". Also having enemies walk right past your AI teammates, who are busy doing nothing, to shoot you in the back over. And over. And over. Gets really frustrating after a while. Two is so much better. You're in for a treat.
[EDIT] on reflection the one mechanic that probably feels off to a modern player in Perfect Dark, and indeed in many older shooters, is having thrown weapons treated like ordinary weapons, rather than a secondary attack with a separate set of items to cycle through. That is, you have to switch away from your gun to arm a thrown weapon, then throw it, then switch back. That's a little clunky. IIRC throwing-is-separate model wasn't popularized until the first Modern Warfare (I've played the first Call of Duty recently and I don't think it was in that, yet—man, talk about one that doesn't seem nearly as good on re-visiting)