Lots of buses from all over BG to Istanbul. Or the train. Get the sleeper and you arrive in the centre of Istanbul mid-morning.
To get back to the topic, MAGURA near Belogradchick is truly amazing. It is not well known even within Bulgaria. It was used in the past as a centre for the treatment of respiratory disorders, and was obviously an important part of Civil Defence planning in BG at the time of the Cold War.
I found a travel agent in Sofia that was offering day trips and I am pleased that I did it when I was still able to travel. Read up before you go - not forgetting the archaeology.