Everything is possible, whether its economically viable is another matter!
I am not too familiar with the differences, but the obvious one will be steering rack and wheel. The clutch system is also different on RHD being a (notorious) hydraulic system. The LHD is cable operated, so IF you could route a cable round from the RHD clutch pedal and it worked without too much stress, then you might not have to many issues, but the clutch sits almost directly infront of LHD clutch pedal, so for that its a simple straight cable, and I don't think Fiat would have gone to the expense of desiging a whole different (complicated) system if they could have just routed a cable. You will need to see how the brake works, often they are just a bar that goes from the RHD pedal to where the LHD pedal would be - which is right in front of the master cylinder.
I am not sure how much of the under-dash electrics are different, if all the fuse boxes etc are in the same place, then its mainly swapping the main dash components (like airbags), but if they're in different places you either need to work around it, or you have one heck of a wiring job on your hands.
I think a committed enthusiast could potentially do a lot of it themselves. It would be insanely expensive to pay a garage, and you would need to find someone a bit specialist to take on such a big and complicated job.
I would suggest researching thoroughly first - take the dash panels off both cars and wee whats different and whats the same, so you have an idea of how much there is to do before embarking on such a project. Removing the dash is fairly easy, but if you can't work out how to do it yourself, you won't be able to do the rest of the project, as most of it will be you working it out as you go along!
When you say unhealthy, I assume you mean rusted - if the structure of the car isn't condemned by rust, then it's likely easier to replace whatever is wrong with the RHD car - a like-for-like engine swap is not too complicated and plenty of garages would take it on and it wouldn't be that expensive - would only take a day or 2. Even if you have rust, replacing floor panels etc again may not be too expensive. You could also consider just buying a newer Multipla, if you have a mk1 and want that look, the front end is all bolt-on swappable, the rear end the rear quarter panel needs swapping, but a competent welder could do that.