Why was Grey Fox on Shadow Moses island? And how did he get there?

Grey Fox was not part of Liquid's plan and it seems highly unlikely that Ocelot or the Patriots wanted him there. (I could be very wrong in that though)

He kind of just appears after the battle woth Ocelot without any explanation on how he got there.

Could Naomi have arranged it knowing that her brother wanted a fight to the death with Snake?

Or have I just missed out on something?

Fox was there to have his final battle with Snake.
Which, according to Naomi, was his last wish.

Of course that's what he wanted, but the details are hazy on how he got there. Was it an arrangement from Naomi? Did Foxhound bring him with them to Shadow Moses as he was a Foxhound experiment?

Does it really matter?

He's there, that's all that matters. If you get bogged down in the logistical details of MGS then you are never going to be satisfied.

For example: How did the Ninja drop from the middle of the roof into the Conference Room when its very clearly shown that the room has no skylight nor any walkways?

Or: How did Snake swallow that pack of cigarettes when the pack is clearly larger than his oesophagus?

All that matters is that Gray Fox was there.

I want to know, How old was Gray Fox at the time of Shadow Moses. Wasn't Gray Fox a teenager when he met Big Boss.

he was in his mid/late 40s.

I think when BB and Fox first met was in Mozambique, years before (not sure how many) the events of Portable Ops when Frank was just a little boy, but BB defeated him and sent him to a correctional facility were they trained him to be the perfect soldier, but BB was unknowingly about this untell MPO or something like that.

Gray Fox got there by his own without no one knowing it.
His final wish was to have a last death match with Solid Snake and that's probably why he saved him when Ocelot tried to fire at him and why he is in Shadow Moses in the first place.
Naomi didn't knew anything about Gray Fox, why he went there or had to do anything with his arrival.

I wonder why Ocelot and Gray Fox don't recognise each other in MGS, afterall they were both in FOXHOUND, high up officers serving under Big Boss.

Who said they didn't? And Grey Fox had a mask on so Ocelot would have some difficulty seeing who he was.

Didn't they?

Baker is neither dead nor dying, Snake is perfectly healthy and there's no one else there. I'd say he realized who it was, due to him being knowledgeable on the Patriots and their activities; Gray Fox being one of their main projects and who has a knack for dying but being "reborn" (Zanzibar Land, the fire in the lab). At least that's the way I've always interpreted it.

He's talking about Grey Fox, so he knows who he is. He later acts as he doesn't know him around Liquid, in the torture room, so no wonder people might get mixed up. It's a little hint that Ocelot isn't who we think he is.