the piston is to be pushed out from the caliper with compressed air a small pump may work - but not the long bicycle one - but the foot pump that can pump a cars tire. BE VERY VERY CAREFUL as the piston will shoot with great force out of its bore, if your fingers are in between the caliper and piston you will be breaking some bones.
If you do wish to service your brakes - smooth out the sliders and grease them and apply anti squeal to the pad backing plates, Install new rubber brake pipes and flush the system. Many sticky calipers are solved by new rubber pipes, it becomes smaller internally which inhibits back flow (caliper retraction)
If you are with a stuck caliper, and are game to remove the piston, be sure to save the dust boot of it if you cannot find a new one with a kit. The piston surface can be buffed with a cotton high speed buffer to a shine that is very very slick. When installing back always use a new seal and wet with new brake fluid first, Use silicone grease on the dust cover seal and pop it into place on the caliper and piston, ensuring it seats fully, use stiff plastic tools to ensure a good fit - metal tools can tear it. All the internally rusty calipers Ive seen in Pakistan were beyond repair, the pistons and their bores were pitted. Best to get replacements.