With Ethiopian Airlines you can fly daily from Addis Abeba to Axum. The flights are like a bus, they land in each destination. From Addis you'll stop in Bahar Dar or Mekele and have a duration of 2 hours and cost about 35 euro on way.
Flights are not expensive but check before you flight from your country because sometimes if you buy the tickets combine it with the international flight, is cheaper
Travel tips
You can also reach Axum by bus in 4 or 5 days from Addis Abeba. Making a tour and have also stops in Bahar Dar, Gondar, Simien Mountains and Lalibela will take at least 14 days.
You can always walk but if you preffer toget some bikes or
go around in Aksum and visit the ancient sites, it's nice to hire a bike for 6 birr per hour.
Local people use also camels for transport, maybe you can convince them to hire one!!
