If you need to find him, the Time Traveler seems to hang at the Community Living Center, usually on the upper floors. The first opportunity I recommend you do is Buried in the Sand as it offers some monetary reward to help you with the others and is the one that costs least to do (one requires you to buy a Plumbot). You can either do the others until it comes up or repeatedly ask for jobs and cancel until it is offered. You don't necessarily have to do the chain, but you'll get more Simoleon rewards and Advanced Technology Skill points if you complete it. The Time Traveler will send you off to the Wasteland to excavate more sites and collect Mysterious Fragments. With your first Excavation, you should have attained at least one mysterious fragment. After finding more, you will be stricken with Wasteland Pox and need to visit the hospital. I skipped doing this until I had Excavated a few more piles to get all four parts, to ensure that if I did not get them they would respawn later. It's unlikely you even need the Opportunity to open the door, but the experience makes the process worth it, imo. Though you are likely to get the fragments, skip assembling the key until the Time Traveler gives you that Opportunity.