In the movie (and largely due to the need for tight editing) you could be mistaken for thinking that all they did was switch out the battery. According to the film's Junior Novelisation, however, it's clear that the boys spent some considerable time getting the jeep back into a workable condition, we just didn't see it. This presumably includes re-inflating the tyres and turning the engine over manually a few times to make sure that it's not seized.
You may also want to note that the fuel for the Jurassic Park jeep came from a Jurassic World vehicle, not from the original jeep. Presumably whoever stored it in the garage would have taken the time to drain off the fuel to prevent clogging.
“A 1992 Jeep Wrangler Sahara, beige,” Gray said.
Zach popped the hood and looked at the engine. “You remember all that
stuff from when Dad fixed up Grandpa’s Malibu?”
They went back outside to the crashed Jurassic World vehicle, salvaged
the battery, the spark plugs, and a can of gasoline attached to the
back. They worked fast, on edge, listening for the slightest rustle of
leaves. Zach heard the roar of a dinosaur in the distance.
“I mean, I know for sure that thing is definitely not out there,” Zach said. “Totally safe.” He handed his brother the car battery. “Here. Take this. You’re stronger than me.”
Gray smiled. They lugged the car parts back into the garage and started working on the old Jeep.
In the garage attached to the old Jurassic Park visitors’ center, Zach finished connecting a cable to the battery. Gray sat in the driver’s seat, peering over the wheel.
“Okay,” Zach called. “Turn it over!”
Gray turned the key. The Jeep chugged, coughed…and started!
For what it's worth, in the original novelisation, the equipment sheds contained many useful items including fuel, oil and spare tyres. It's clear that the staff were doing their own servicing (due to the remoteness of the island) so there's probably spares of everything you'd need to get an old jeep running.
Grant stumbled deep in the gloomy recesses of the building. He pushed
past five-gallon containers of herbicide, tree-pruning equipment,
spare tires for a Jeep, coils of cyclone fencing, hundred-pound
fertilizer bags, stacks of brown ceramic insulators, empty motor-oil
cans, work lights and cables.