Ubud is about 25 km north of Kuta and the international airport, a journey that takes 45 minutes to 1.5 hours depending on Bali's notoriously unpredictable traffic. There is no public bus service between the airport and Ubud. The standard options are a metered taxi from the airport (agree a fixed price at the official taxi desk in the arrivals hall), a private transfer booked through your accommodation, or a car rental with driver arranged in advance.
Kura-Kura tourist buses run between the main tourist areas but do not serve Ubud directly. Once in Ubud, the town itself is walkable. The surrounding countryside is best explored by rented scooter or bicycle, or by hiring a car and driver for a day.