Hi,
There was a thread over in the 500 Classic section of this forum that advised that the correct adjustment for the gear lever was for it to be vertical when in neutral. (the 126 uses the same arrangement for the gear linkage).
But having said that, if all the other gears, especially 3rd gear are being selected correctly and not jumping out of engagement, your gear lever adjustment is probably ok.
Before removing the gearbox, there's 2 things you might check:-
1) Under the car where the gear linkage rod attaches to the gearbox, there's a bonded rubber/metal connection - check if this has started to separate or if the rubber has gone soggy from any oil leakage.
2) When 1st gear is selected, the gear lever inside the car rotates this gear linkage rod clockwise (when looking back towards the gearbox) at the gearbox and pulls it forward (towards the front of the car) - check if this is happening correctly i.e. lie under the car looking back towards the gearbox, have someone select 1st gear, you should see the gear linkage rod rotate clockwise and then be pulled forwards. You might check if the gear linkage rod is moving as far forward as it can go - maybe disconnect the rubber/metal connection and see if the gear selector rod sticking out of the gearbox tail-housing can be extended some more.
If neither of these things are at fault, it seems likely that you'll have to remove the gearbox for examination.
It might also be useful to try posting this thread in the 500 Classic section, many of the 500 owners have worked on and fitted the 126 gearbox into their 500's so have considerable experience of these gearboxes.
Hth,
Al.