Restaurant name = Hanabishi
You will recognize it from the purple noren curtains and the line of people outside.

http://r.gnavi.co.jp/Mg/50/3006050.html

Bento (not vegetarian) = 2,940yen
EDITED - THE BENTO here are not veg. (My memory let me down before).

The shojin dishes ARE buddhist vegetarian:
Shojin kaiseki lunch set = 2,100yen
Full on shojin kaiseki sets = 5,000 to 10,500yen

It's worth going into the store that sells traditional Japanese sweets, which is on the same road - maybe 50 meters closer to the oku-no-in.
I don't recall the name, but it's the one with all the cakes displayed on wooden box-tray things. There are some tables inside where you can take a cup of hoji tea and a sesame cake or something. Nice.