It'll work. But - you won't have the range of a normally tuned guitar, it'll be smaller.
You'll have to work out new fingerings for each and every chord. On guitar, it's tuned like it is to make as many chord shapes as possible accessible. Otherwise it probably would be tuned differently!There are 5 frets between strings and one reason is most of us have 4 fingers, so we have all the chromatics on one hand, before we need the next string open.
Writing out music isn't going to be a problem - each note is where each note is, although the names of the notes are going to be a little confusing - but that's all. Most people tune with a guitar tuner, so sharp names will most likely be favoured. if not, then there's no problem to tune as you say, with those note names. Or, using the 4th and 5th fret method, only in this case, it'll be 3rd and 4th fret method.
I'm interested as to the reasoning behind why this may have advantages over normal tuning.