Axum has a small airport, Axum Airport (AXU), with daily flights to and from Addis Ababa on Ethiopian Airlines. The flight takes about 90 minutes and is by far the most practical way to arrive; overland routes from Addis Ababa are long (roughly 14 hours of driving) and were affected by the Tigray conflict of the early 2020s. Check current road security before attempting any overland journey.
From the north, Axum is connected by road to Adwa (30 km) and Adigrat (130 km). Bus services run along the main routes but are slow and basic. Most independent travellers fly and use local taxis or tuk-tuks to get around town once there.
Note that the security situation in Tigray has varied significantly in recent years. Verify travel advisories from your government before booking.