What are some nearby airports?
Austin Bergstrom International Airport (AUS).
Aloft Austin at The Domain, 11601 Domain Drive, Austin TX 78758
There are several other accommodations available in the area offering a variety of lodging types and prices. You may want to conduct a search online for options available for your reservations. It's preferable to be within walking distance of the training location to avoid traffic.
Depends. A taxi from the airport to your hotel should suffice, and shuttle services from the hotel are often provided—check with your hotel.
Do I need to fly in the day before and leave the night after?
This primarily depends on if you are able to arrive early enough on the day of the class to meet the scheduled start time. Most people tend to fly in the night before a training so they can check into their hotel. It is very possible to catch a flight out the same night as the final day of training, as long as you are conscious of the 5:30 PM end time.