I've used a heavy bar with a piece of wood to pry on. Dig around one edge and stick one end of the bar under the slab. Have some help and when it's raised up shovel fill under it. Then go to the other side and do the same. In about a year you will have to do it again because it will settle, but after that you might be okay for a while.
I've done this to the city sidewalk (4'x4') when they wanted to charge me to replace sections that had sunk.
If you can't find a bar, you can use an old piece of steel pipe...say something about 1 1/2 -2".

"...shovel fill under it."

Use pea gravel.

What about a company that does mud jacking? Basically they drill a series of 3/4"-" holes in the concrete that has sunk and use special equipment to force a concrete slurry under the slab to lift it.

Mud jacking is not worth it for small enough sections to lift mechanically like a sidewalk or walkway.
It works well for a large patio slab or driveway sections.