Patna's Jay Prakash Narayan Airport (PAT) sits inside the city limits and has frequent flights from Delhi, Mumbai, Kolkata, Bangalore, and Hyderabad, plus seasonal connections to Dubai and Kathmandu. It is small and dated, but functional.
The bigger gateway in practice is Patna Junction, one of the busiest railway stations in eastern India, with overnight trains from Delhi (about 14 hours), Kolkata (8 hours), and most other major cities. The Rajdhani and Tejas services are comfortable; book a few weeks ahead in season. Long-distance buses leave from Mithapur bus terminal for destinations across Bihar, including Bodh Gaya.