BSG
Can Apollo only use his ability on his own turn?
No, he can use it whenever he plays an XO; however, in order to use it on someone else's turn, he would need to find a way to play an XO. He can't be XO'd to do it, as you can only have one XO per turn. Old Apollo could get a free action on someone else's turn with AVP, but of course Old Apollo can't use Newpollo's ability.
One way at least that I know of for this to work is if someone plays the State of Emergency-6. That gives everyone a move or an action, and doesn't count as an XO. Apollo could use his action to XO someone and then activate his own location too, as long as no one else plays an XO first.
A quick wrap-up of the other players:
Leoben can only use his ability on his own turn, as it takes up his Movement phase, which only occurs on one's own turn.
Baltarnate can use his ability whenever he has an action and another player in his location to steal from. I think he could even use it twice in a turn on an XO or such like.
Hoshi can only use his ability on his own turn.
New Helo can use his ability whenever he plays a Launch Scout. He could use it more than once in a turn if he scouted twice on an XO.
Cottle, likewise, can use his ability whenever he has an action, and can use it twice on an XO, as he did last turn.