Stukachov initially comes off as a smarmy kiss-up: "One neighbor, Stukachov, follows us down the corridor, smiling and bobbing his head, asking how many spies my dad has exposed today" (4.1). Just from this, you might not get a good handle on whether he's being sarcastic, or flat out sucking up to Sasha's dad (since lots of people are afraid of him and apparently don't want to get on his bad side).
We soon find out that Stukachov has it a bit rougher than Sasha's family. He and his wife, three small children and mother all live in a tiny room (while Sasha and his dad share a rather large room in comparison. So, there's some sympathy built in for Stukachov. It's quickly dashed, though, when we find out that Stukachov has sold out Sasha's dad and wastes no time moving his family into Sasha's room (leaving him out on the cold street—literally!).
On the whole, he seems like a pretty crummy character. So why does Yelchin throw in some sympathy there? It's almost like Stukachov re-creates the experience of super-paranoia that pervades the book: people are always on the look-out, and even if you like someone or feel sorry for them, it's not worth your life to stick your neck out for them. It's more likely that you'll end up selling them out just to save your own skin, or to better your circumstances—which Stukachov accomplishes quite handily.
And there's also the more practical issue of plain old physical survival. Stukachov has a large-ish family, which was previously squeezed into one small room. He now has much more comfortable digs for his family, which might seem totally worth it given the harsh circumstances we're dealing with here.