Aye, that's basically what he had in D&D terms. At some point, maybe just in the later Ultimate X-Men comics, he became a swordsman as well (at least, judging by what he's like in the video game X-Men Legends 2).
Nightcrawler would have an ability like the spell Spider Climb (or the psionic power Wall Walker, which Psychic Warriors can get, or the Shadow Hand martial stance Dance of the Spider, which Swordsages can get). Plus huge Dexterity, some unarmed combat prowess, and significant teleportive abilities. And a prehensile tail....somehow.
Short-range teleportations didn't put much strain on him, so he could do a lot of them in a short time, but they'd still exhaust him physically and mentally after a few dozen short teleports. He could make only a few longer-range teleports before exhaustion, similar to a Dimension Door or something. The furthest he could teleport was maybe one country or one continent in distance, I don't recall. But it was really painful and exhausting for him to go that far.
Probably the most accurate representation of Nightcrawler would be a Psychic Warrior, multiclassed a bit into the unarmed-combat-focused Swordsage variant, taking the Overchannel, Talented, and Psionic Meditation feats at some point. Probably taking the Dance of the Spider martial stance for his wall-crawling, since it has unlimited use. And learning any powers they can to boost their Dexterity score or bonus.
(my favorite Marvel character!)