It seems I will have a few hours to spend at the landside of Budapest airport. There seems to be a cafe Edit: http://www.bud.hu/english/passengers/shopping_and_gastronomy/restaurants/flocafe-8131.html but I can't find information whether it has power outlets. Although I have Priority Pass, all the lounges are airside and the only landside lounge I was able to find is part of a VIP package and as such it has a rather high price http://cavok.co.hu/pdf/pricelist.pdf and is not included in Priority Pass.
Having spent time at the cafe mentioned, I can tell you that it indeed does have power outlets and comfortable seating.
However, as it is the prominent cafe at check-in; it gets crowded quickly.
There is often a large line at the counter. If you are traveling with a partner, have one reserve a seat for you.