The answer is yes - Fiat have done all the work for you - just find yourself an old NA 1300 Uno and nick all the bits you need. If you aren't going for the turbo then this is probably even easier.
There is a whole range of old Fiats that used variants of the same SOHC engine that were injected - any of them will do...
There are of course X1/9s that were factory fitted with injection systems.
Thing is - don't go expecting too much by way of economy, the injection system is better, of that there is no doubt but the improvements in economy come more from the camshaft profile. The standard european X1/9 camshafts are closer to what most other european manufacturers would class as fast road profiles. There was a time (many years ago) when the usual camshaft suppliers were actually downgrading the X1/9 profile without realising it.
What you need to do is move peak torque down the rev range so the engine doesn't have to rev so hard, you can then cruise around without much throttle and still get reasonable response. The downside is a loss of top speed and performance in the upper part of the rev range.
The better alternative is to just make your car lighter - it does wonders for economy!