My fiat 500 hit 235 or 240 km/h about 146 mph but it's tuned for race and not for drag, i use a 6 fully custom gearbox, but i've 269 HP not 225 and a weight of 853.
Yes you have to buy it, all your horespower upgrades are meant to make your car faster but you haven`t change its topspeed so far with a regular gearbox! Even a short 6 gearbox won`t do that much in many cars.
If you hit max BHP for a car I always recommend a adjustable gearbox.