holborn,
Besides the Cathedral as mentioned by Colin you can get your first stamps anywhere in Burgos. The tourist office, bars, hotels, albergues, etc. all have one; there is no 'official' stamp.
Finding an bunk after 6pm depends on what might be described as "the luck of the draw".
You can only reserve a bunk in advance at private albergues; If you have a reservation you should be assured of a bed. Check the Burgos albergue listings here
http://www.caminodesantiago.me/camino-f ... to-burgos/ to see if any take reservations.
At municipal albergues it is first come first taken. However some of these places have or know of 'hidden' backups, Ie another floor or adjacent out-building. Often hospiteleros will call around to help find you another affordable space.
Thus finding a bunk in the busy season can be difficult. That's one reason I have always gone in autumn and winter. It may be cold but there are plenty of bunks!
Buen Camino,
Margaret Meredith