Glen Jodan Liversidge was wanted by police for the break-in at Mex Bar on Albion Court on May 17 when he was caught committing another burglary with an accomplice.
Liversidge, 34, of Saville Street, appeared at Leeds Magistrates' Court this week.
He had been arrested on Sunday evening after a homeowner on Portobello Road called the police when they spotted two men burgling the shed.
They stole a pedal cycle and a child's scooter among other items worth a total of £160, but officers caught up with them a short time later.
One was immediately arrested while the other fled. He too was arrested a short time later.
Liversidge and 28-year-old Reece McGarry were charged with a non-dwelling burglary.
Admitting the bar break-in and the shed burglary, Liversidge was handed a 40-week sentence and told to pay £128 costs.
McGarry, of Phillip Garth, Outwood admitted the shed burglary, and a second charge of stealing razors worth £65 from Boots on Kirkgate.
He was jailed for 20 weeks and told to pay £128 costs.