You can play them emulated on an arcade cabinet.
I do.
As for some good ones to try, here's a short list:
Metal Slug 1, 2, X, 3 (4 is OK, but not great) - military action at its best
Waku Waku 7 - zany, happy, nutty fighting
Last Blade 1, 2 - classy 2D weapons fighters
Puzzle Bobble 1, 2 - fun action puzzlers
There are so many more, but those are some of my favorites. In fact I was just playing Metal Slug 3 on my cabinet this afternoon.
It's even more fun with two players. And that's a big part of it - NeoGeo games IMO are best played with another person. They worked really hard to make the games extra fun with two players, simply because that means more coins being inserted.