There's no good answer, unfortunately. There's a troubling correspondence of Germanic p with Irish and Slavic p, so if nothing else, we can at least be certain that the word was not "around for a very long time; accumulating all the sound shifts." If it had been, we wouldn't see this particular set of correspondences. Germanic and Slavic p do not correspond in inherited words. Rather, Grimm's law gives Germanic f corresponding with p (as in English fart and Russian perdet'). I'm no expert on Celtic, but Irish at least generally lost PIE *p (as in athair 'father'), so the p in those forms is also anomalous. This means we're dealing with a relatively recent formation that is definitely not PIE. The Slavic forms with /o/ are consistent with it being a borrowing from Germanic into Slavic, predating/concurrent with the split of *å into *o and *a, in the 7th or 8th centuries, roughly.

It's hard to say when the word showed up in Germanic, but the OED has a citation from around 700. Since it must have shown up after Grimm's Law stopped operating, we have a window of a millenium and a half, give or take a century. (One might object that this could reflect some old PIE form with *b, which by Grimm's law would give Germanic p, but PIE *b is vanishingly rare in roots, and it's quite suspicious that we don't see cognates of the word attested in any other branch.)

I can't comment on the presence of apes in ancient Europe, but I can say a bit about the origin of the word. It is from the Proto-IndoEuropean root abō(n) (hebō-), probably originally meaning 'water-demon' (root ab- 'water')(cf. Adsoqiation 2007 entry for the same). It's meaning as 'monkey' is probably a neologism, either already in PIE or from very early Celtic. If any PIE daughter language is the origin, it is probably Celtic first (cf Orel Germanic Etymology 2003). Adsoquiation 2007 assumes a monkey was introduced at some early point and was given a nickname derived from abō 'water demon.' Personally I love this etymology. Direct descendants include Germanic apōn, Old Norse api, Old English apa, Old Saxon apo, Old High German affo, and the ones OP lists above as well.

In Sanskrit, kapí, and also Greek κῆπος are a little problematic. Greek shows variants κῆβος / κείπος / κῆφος, the latter probably on analogy with Latin cephus / caepus, of unknown origin. The initial /k/ has led to speculation of a relation to (or analogy with) Hebrew qōf, Old Egytpian qefi ('ape of the land of Punt'). Of these, Egyptian is probably the best fit as a source of a loan-word.

In Latin (and thence Romance), the most common word for monkey is simia, borrowed from Greek σῖμος (adj) 'flat-nosed, snub-nosed; impudent (metaphor)'. This Greek word is strange in itself, also of unknown origin, though attempts have been made to tie it to the PIE root tiH2-mo 'hit, struck.' Another Greek word, πιθηκος 'long-tailed monkey,' is also of unknown source. Beekes (Etymological Dictionary of Greek 2010) has some references under κῆπος.

Slavic opica is probably a late derivation from Germanic (via Orel 2003).