Gearbox jumps from 4th to 3rd

Sounds like the cable is loose or an cable adjuster needs adjusting, check both in the engine bay and under the car, something is loose.

Sounds like you have worn gearbox selector linkages and I suggest that you get someone to take a look at them PRONTO.

you have to update the program to adapt with the new gearbox, by visiting the nearest agency or specialized workshop.

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check oil level in gearbox, if auto check fault memory, will go into emergency mode and stay in 3rd if there is a fault. gear position display will all light up.

I think this car has a torque converter lock up. sometimes they freeze up when the the pcm makes 1:1 to the motor and transmission. this could be the shifting problem.

did you change your clutch when you changed your gearbox??? also check the clutch master cylinder.

You need to give the car a scan, preferably with VCDS (or VAS at a dealer). If limp mode is activating (lit up gear selections), then it will be storing codes.
The fact that it works to a degree could be good news, the gear box has likely not failed. Could be the F125 selector switch, which is a common fail at this age.

balls

After a jump start because of no electrical power, the car will run rough for several days (expected) while it adjusts to your driving habits. I'm not sure about the gears, but I'd try to drive with it like that for a few days and see if it returns to normal.