You can spend a night at the monastery. There are dormitories and private rooms aimed primarily for people on Orthodox Christian pilgrimage visit to the monastery.
Booking can be a bit complicated as it can be done only by telephone so you would need to contact the monk brotherhood of the monastery directly.
An easier option is to make a reservation at hotel Tsarev
Vrah which is owned by the Rila Monastery.
The rates are reasonable and it is located just next to the monastery.