My son is an Arsenal fan (in his defence he does have a gills ST and is slowly migrating over to gills).
What that means is that I have 2 Arsenal Red Memberships and one Junior Gunners ticket which I use to get him the odd ticket to games. Arsenal sell out most games, but do guarantee a certain amount of tickets to Red Members. It's about 5000 tickets every game go on sale to red members a month before the game on their website. You just have to be clued up and lucky in the online queue on the day they are sold.
Some games are easier to come by than others, but by way of example I got tickets this way to Arsenal vs Liverpool, Newcastle, Swansea, Galatasary for him last year.
I'd be more than happy to try to get some for you if you let me know the game you fancy, and if I'm not using them (probably won't be).
They also almost always have tickets in the Junior gunners sections at this point when these go on sale - the only condition being that you book your ticket with a child's. So what i've done for mates who are arsenal fans is book 3 tickets (2 adults and 1 child) and he just pays for all 3 but only 2 adults actualy go. The child's ticket is only about 15 so not too much extra, and maybe worth it if you otherwise wouldn't have the chance.
I also have a few spurs supporting mates who are on their membership schemes who might be able to have a go if they are not going to those games already.