Any firearm will go through multiple walls in an apartment, and a shotgun would just result in a ton more projectiles going everywhere, increasing the chances of injuring or killing someone if you miss, especially since hitting the first wall will usually make the shot scatter pretty widely. If you wouldn't use a .32 ACP submachine gun, you probably shouldn't use a shotgun with buckshot either.
Your best bet would be to try and use the apartment's layout if at all possible, and put some kind of backstop, like a completely filled bookshelf, behind any "choke point" approaching the master bedroom. A couple layers of big, thick books taped together, like phone books or JC Penney's catalogs, would work
great, and could be hidden behind a close-backed shelf or something, so they're not unsightly. Do two layers and overlap them so there are no gaps a bullet could slip through. Paper is surprisingly good at stopping pistol JHPs. FMJ ball will whizz right through
several inches of paper, but in my limited experience,
none of the pistol hollowpoints I tested made it more than halfway through even a single JC Penney's catalog. I shot it with a couple types of .22 LR, .357 magnum 158 gr SJHP, and a few different 180 gr .40 S&W JHPs. They all expanded in the paper and stopped. Whichever gun you end up deciding to use, take a few phone books or catalogs to the range and try shooting them, so you know about what thickness you'd need.
I'd stick with a handgun, or maybe a pistol caliber carbine. Use a good quality hollowpoint, pre-fragmented ammo is a gimmick. If your handgun is a revolver, or you get a revolver caliber lever-action carbine, use semi-jacketed hollowpoints, as those will usually penetrate building materials less. 12 gauge shotgun with reduced recoil 00 buck would also work, but wouldn't be my first choice, especially if you're new to shotguns. The manual of arms for any manual action (pump, lever, etc.) takes a bit more getting used to, so you and your wife would want to practice more at first, compared to a semi-auto.
Spread on shotguns is overrated (until the shot hits a wall, anyway), so you really can't count on buckshot making it any easier to hit. At apartment distances, you can expect the shot to spread less than 6". Probably less than 4".
And check out the box o' truth for comparative penetration of stuff.
http://www.theboxotruth.com/
If I were in your position, I'd probably go for either a Kel-Tec or Hi-Point pistol caliber carbine (as budget dictates) in the same caliber as the pistol.
