Yes the lot is attached to and managed by the hotel. Heck sometimes they even leave the gate open for some reason, but they do require a hang tag for parking. Ask for room 301 or 401 to get a slightly larger room with great view of the airport.

Be careful when pulling in to the hotel because the road from Presidential Blvd to the hotel is not designed for anything but the smallest of cars. Instead take the Spirit of Texas Drive exit from Hwy 71 and turn left just before the newish cell phone lot gas station thing on Spirit of Austin Ln.