Why aren't you managing him sensibly?
If he goes and pisses upstairs when you're out theres a good chance he is unhappy being left (scent marking when alone suggests a degree of anxiety, he's scent marking to make himself feel more secure).
I actually would not neuter immediately, if he is scent marking due to anxiety/stress then removing testosterone is likely to cause a sudden drop in confidence which could increase the scent marking.
I would confine to a pen when you're out - check he is actually ok being left via camera and test absences where you're not really gone, you are watching the camera from the garden or just down the street. If he IS ok, he will relax, sprawl out, play with toys, work on a filled kong or chew. If he is not then he's not going to do those things he's more likely to scrabble, dig, climb out, bark, howl, wee/poo.
Asides from pissing when you're out, what else is he doing that you hate - what does he do all day?
You say he has a walk and he's fed - wheres the training, enrichment, games, activities. Training isn't something you do for a bit when they're young then stop... it's something you do throughout the day, every day, for life!