Dating Fuinur is difficult. He appears in the narrative after Sauron returned to Mordor, but before the first capture of Minas Ithil, which in principle gives us dates somewhere between II 3320 and II 3429. The passage describing Fuinur, however, does not seem to follow in strict sequence with the events described around it (it mentions events that must predate the Downfall of Númenor, which took place in II 3319). On this basis, it's hard to say anything definitive about Fuinur's dating, other than placing him somewhere in the closing years of the Second Age.