Exactly. Klingon first contact with the Federation probably went something like this:
Ambassador Krude: "Your pitiful Coalition of Planets makes lousy booze!"
Ambassador Skon: "Actually, Your Eminence, it just became the United Federation of Planets - they have entered the last signature now."
Ambassador Krude: "Ahh. Your pitiful United Federation of Planets makes lousy booze, then! But this signing whachamacallem banquet is pretty cool otherwise. What do you call these sturdy and spicy delicacies?"
Ambassador Skon: "We call them candles, Your Eminence. And chandeliers."
Picard says that "we" first contacted Klingons "centuries" prior to the 2360s, and that this contact was disastrous but apparently would have been less so had there been initial clandestine surveillance before open contact. Picard may of course be lying, because he recounts the story in order to convince the aliens of the week that covert surveillance is a good thing. But he'd have little reason to lie about the date, even if he lies about the role of surveillance.
Picard's "we" would probably refer to some species that later became a UFP member, but "centuries ago" might just barely allow for contact between Klingons and the newly founded UFP as well. Of course, several UFP members would already have contacted Klingons prior to that founding event, including Picard's own species, so this would make little sense - except again if we assume Picard is telling a pleasant and useful lie and twisting even the factual bits past breaking point to accommodate the greater lie.
As for when Klingons went starfaring, nobody really knows. Might have been long before Kahless, or long after Kahless. Just because Kahless is depicted as a medieval or even early iron age sort of hero doesn't mean he wouldn't have had a starship; he's being depicted in legend only, after all. Plus there's the bit about the Klingons being enslaved by starfaring aliens called the Hur'Q; Klingons might have traveled to the stars in Hur'Q ships, possibly as galley slaves of some sort, and made their name there well before they gained the ability to build their own starships.
Or for all we know, Klingons and Vulcans met each other a few hundred years before ENT, when Vulcan explorers first reached Qo'noS. Vulcans forgot to do the preliminary survey and ended up having the secret of interstellar travel stolen from them by the Klingons. Klingons became an interstellar menace, while Vulcans subsequently adopted their Prime Directive, and imposed it on humans and eventually the Federation as well.
Timo Saloniemi