"Faulty actuator pump" sounds like a confused conclusion. To give you a crash course in my limited understanding of Selespeed... there are these parts:
- Pump. As David C says, this needs to run when you open the door. It should run again when you turn the ignition on, then at intervals thereafter to maintain system pressure at 48-50 bar. A large fuse and a relay provide power. Obviously, a good battery is important.
- Actuator. This is the block on top of the gearbox; has hydraulic rams that operate the gear selectors, with positional feedback, and another ram (cylinder) to operate the clutch. The clutch also has positional feedback.
- Accumulator. A canister attached to the actuator, which looks vaguely like a Hydragas displacer unit or Citroen sphere, if either of those mean anything to you
It has nitrogen gas under pressure, and its purpose is to 'store' hydraulic pressure (by compressing the gas spring). If it is faulty, hydraulic pressure will be erratic and the main symptom is apparently that the pump runs for only a few seconds rather than seven seconds or so when you first open the door. If pressure is too high (possibly as a result of the failed accumulator) the gearbox may switch into neutral when driving. If pressure is too low, you could get the symptoms you have (as David C says, maybe there is no fluid).
- Brake pedal switch. Aside from the obvious functions of brake lights, this provides a signal to the ABS and cruise control systems - but also an input to the Selespeed. I think that in a car that suddenly fails to allow anything, if the pump is running, the brake pedal switch would be my next suggestion. You didn't say whether you got the engine started, but that would be possible if you left the car parked in neutral.
- Speed sensor. This is attached to the gearbox and helps the Selespeed to 'know' when the clutch is engaged. I don't know the symptoms of a failure for this part.
- If positional feedback of the actuator fails, you still get 1st and 2nd gears.
- If the steering wheel paddles or the lever fails, you get CITY mode.
- At least you have taken it to someone who has electronic diagnosis; the amount of information the ECU logs is staggering. It knows the number of times each gear has been selected ('inserted') or not selected (which helps to diagnose mechanical problems), distance covered in each gear, fluid pressure, clutch slippage, etc. It can also be recalibrated and tested thoroughly, and any faults in the past (however momentary) will be there to see. E.g. a loss of the road speed sensor pulse, which would cause confused gear selection.
- You mentioned the question of what other work might be necessary. The clutch should be checked/adjusted (the pushrod is threaded for adjustment) using the value given by the diagnosis equipment. The main calibration procedure is called an EOL (End Of Line) calibration, and you would also expect that to be done while the system is connected to the diagnosis equipment. It goes without saying that other faults past/present can be identified and remedied to reduce (but never eliminate...) the chance of breakdowns. I think that a regular session on diagnostics equipment is essential for any modern car. Fortunately you can buy software, such as AlfaDiag, to check it yourself. In the meantime, console yourself that the attention would be required sooner or later; yours is after 120 miles - would you rather have had it at 1200 or 12,000 miles? Maybe you'll forgive it in the long term?
Hope this helps in some way, even if it is only to understand what you are next told by the Alfa dealer. Otherwise it's time to hand over to our tame Selespeed expert. Some say he keeps a log of failed insertions (none, of course), and that his fingers are coated in threadlocking compound before being tightened to 40 Newton metres. All we know is that he's called Smaky...
EDIT: Well, he posted while I was writing that
EDIT2: Forgot to mention - like Smaky, I would rather work on a Selespeed than on a conventional auto. The Selespeed is much simpler in concept, as it's a standard manual gearbox and standard clutch. My 156 Selespeed has logged over 230,000km without missed gears (except 1st, less than 1% of the time), no record of the clutch slipping or overheating, and tens of thousands of gear 'insertions'. I hope you're not put off by your experience so far.