Its possible to enter Bulgaria if you are 'overlanding'.

You can arrive in Bulgaria by train or bus from any of its surrounding countries

Trains in Bulgaria are very cheap and an overnight train will give you a bed in with the price.

For flights to Sofia, use www.skyscanner.net, www.wizzair.com or www.easyjet.com

It is also possible to fly into Istanbul in Turkey and travel by bus to Bourgas then by bus to Sunny Beach.