I believe Spock would be open about where/when he is from, since he is obviously in an Alternate Reality to which that future generally does not apply.
Since he is hanging around Vulcans, he may take on the name of Selek, or keep the name Spock with which he is familiar.
His goal is primarily to help the Vulcans establish a new colony, and from the perspective of everyone who finds out his story, he may dispense the occasional useful bit of info, but his future knowledge would likely be incomplete.
He's likely to be well protected by the Vulcan council or the Federation, but he actually still has little to really hide.
If one doesn't know the questions to ask him, one cannot glean relevent to ones future in any meaningful way.
Q: Am I going to die? if so, how?
A: I do not know. The future is unpredictable to me in this Alternate Reality.
Q: What lies out there for us to find?
A: Here is what happens, and what I know. I hope it helps with dealing with the issue.
Either way, I strongly suspect he's not going to be called upon in future movies, except perhaps as an occasional mentor to young Spock, and that possibility is interesting if remote.