Painting should be fairly straight forward. I'm assuming that it is just woodstain and not varnished... either way, give the surface a light sanding to give a good key, remove any loose material, ideally give it a wipe over with some Methylated Spirits. I would then give it a coat of Zinsser B.I.N. Once that is dry give it another light sanding and you're ready to paint.
That design looks a bit prone to 'lateral stress', so making it in to a cupboard would only help. I would fix some board – 12mm MDF should be fine or 18mm off the same sheet as doors – to the back and sides, and to the front face of the two corner uprights, flush with the 'inside' face... if you see what I mean.
You then need some doors. Simplest would be two 'surface mounted' doors – these I would do in 18mm MDF – using concealed hinges
(the type you normally get in kitchen cupboards).
You need a special drill bit for concealed hinges which you can buy for about £5. Or maybe 'Easy Mount
' would be better. I've never used Easy Mount, so can't say anything about their quality etc.
Hope that helps.