For AL play, the same character can only participate in any single adventure one time. You are not allowed to play an adventure a second time, even if you take no XP or rewards. You can replay it with another character though.
That said, the Moonshae Preview adventures appear to be 4 separate mini-adventures. I can't speak to how they are specifically set up, but previous intro adventures of this kind involved 4 1hr mini-adventures that were scheduled (and often took) 2 hours to play through each one. If these are set up the same way, your new character can indeed participate in each of the four mini-adventures. When you play your first game, you'll note which mini-adventure it was. The next time you play, your DM (or mustering official) will ask who has played which parts so that you can sit at a table and play a new part for your character.
As for XP, previously the intro adventures awarded about 100XP per mini-adventure. If that trend holds true, you won't get level 2 only playing one of them.