Hi Denon,
OK, first of all, as far as I understand the Ace did not raise the initial bring-in right? In that case he may have 1. a three flush 2.random high cards 3. small pair in the hole or 4. an ace, which he is slow-playing (and the fifth possibility, which you must not underestimate is that he is an idiot playing any three cards with an Ace in them).
First of all, you have to count the cards which are the same suit of the shown Ace. How many are there on the board? If there are no more than two, then a flush draw is possible, which is bad news for your straight draw. Drawing to a straight may turn out to be very costly in the end if there is a strong flush draw in the board. Small pairs in the hole are not big threats for your hand: If you make your hand, you will most probably win the pot against their likely made hands. If he has random high cards in the whole, that is bad for you, because you need those high cards in case you do not make your hand (that is, you may still win the pot if you catch high cards along the way even if the straight does not materialize). If he is slow playing an Ace in the hole, then you have a big problem. You need an Ace for your straight draw and two of those Aces are in his hand. Moreover, even if you catch a high card along the way, you are a big dog to his made hand.
People in general exaggerates the power of their straight draws in Stud-Hi a lot. You should not do that mistake. I would continue to chase my draw if the player with the Ace does not show much strength or if the other players do not pair their door-cards. If nobody shows much strength and you catch a few high cards to make a two-pair by the river; then you should at least call a bet on the river. If the ace suddenly comes alive in the fifth or sixth street and begin to aggressively re-raise (or if he shows a three-suit and re-raises etc.), then it may be prudent to get out. Straight-draws are not powerhouse hands and therefore you should in general not aim to create huge pots with them. Why? The reason is that if somebody is willing to come for the ride and raises and re-raises and help you to create this huge pot, it means that, most probably your final hand (that is a straight) is already dead.