It will cost a lot of money to turbo your car, and its reliability will drop substantially.
If your serious about this, get a turbo 2.2L motor. It is the same displacement but with turbo-ready components. It has a fully closed deck as opposed to the non-turbo's semi-closed deck, as well as a lower compression ratio. This still won't put you anywhere near the 250 HP you're looking for. For that you're going to need a different turbo, some bigger fuel injectors, a 3 angle valve job with ported heads and intake manifold, under-pulley kit, full 3" exhaust, intake, and an MSD ignition swap. This will put you in the low 200 HP range depending on the condition of the engines internals. If you want to get 250 you need to perform these modifications AND have the engine fully rebuilt with a slight overbore. While your at it, some larger valves would help. The only problem after that would be the transmission. If the car is an automatic transmission you're going to need a separate transmission cooler to cope with the added stress. If it's a manual transmission you're going to need a good clutch, or you'll be replacing clutches about 3-4 times a year. CV shafts will also take a beating.
Honestly, if I were you I would just give the car a couple bolt-on modifications like an intake, an exhaust, MSD ignition, and an underpulley kit then put the rest of the money in the bank. Don't bother with an engine swap. This is your first car, you don't need to go all out yet. Save your money and if the current engine dies out on you then worry about replacing it with a turbo engine. The best part is you can swap all your modifications over to the turbo motor and get decent performance numbers.
The problem with the EJ22 is the HP. They had excellent throttle response and a great torque curve, but their HP numbers weren't great. The non-turbo motor put out about 135 horsepower, and the turbo motor didn't even put out 170 HP. 250 HP is certainly possible, but not very practical on a $4,000 budget using old worn out parts from junkyards and Ebay.