We will be arriving in Jakarta on Sept 29th at 12:10 a.m. and flying to Yogyakarta at 2:20 p.m. Any suggestions on hotels close by the airport to avoid traffic and get to sleep as soon as possible?

The Sheraton Bandara is very close to the airport (about 5 minutes way). It's acceptable and functional with an okay pool and restaurant. They also have a spa.

With that much time you could go into the city and stay at one of the nicer hotels like the Grand Hyatt, with a really nice pool and spa, big breakfast buffet, and nearby shops. You won't get traffic on arrival and as long as you allow ample time, getting to the airport for a domestic flight shouldn't be too onerous.

Curious though why you don't take an earlier flight to Yogya? You could also substitute Solo (Surakarta). Even Semarang if you don't mind the drive to Yogya.

Good question ! Because I hadn't realized that the 12:10 was a.m. when I booked

In your opinion in what order should we visit borobudur and Bromo to then return to Seminyak and what would be the most efficient travel method?

I don't think it makes a difference whether you visit Borobudur or Bromo first as presumably you'll want to fly between Yogya/Solo and Surabaya and between Java and Bali.
Jakarta-Yogyakarta-Surabaya-Denpasar makes the most sense, but since flights are quirky you may do some backtracking. You just have to experiment with the flight schedules and see what suits your purpose best.

You can also drive from Bromo to the Java-Bali ferry then drive down from North Bali to wherever you want in Bali. This is doable but only if you have the time and spirit of adventure.

Ladouca, we are headed to Java in November. We plan to visit Yogya/Solo area and Malang and maybe Bromo. There are flights from Yogya to Surabaya on just one airline, but we have decided to take the Executive Class train instead. Then there are plenty of flights from eastern Java to Bali.