I brought a player up last year after roster expansion and then sent him down to AAA before the next season began. Will his time on ML after roster expansion count toward service time? When can I bring him up this season to not have this season count as a full year for service time? After 20 games or do I have to wait the additonal number of games he was in ML at the end of last year?

Yes, his time in the ML after roster expansion counts.

You can see his precise service time to figure out how long to keep him down by hovering your mouse over the "0" found after "ML Years" on his payer card.

And when it gets to a full 1.000, then his clock starts ticking?

Well technically his clock "ticks" every day he's on the ML roster. But yes, when he reaches 1.000 that means he has used a full pre-arbitration year.

I would note that time on the roster during the playoffs does not count in this calculation.