Is it possible to get an A-lister into a big-budget movie without anybody knowing?

If that movie has no other stars, it's unlikely a studio would want to intentionally minimize the presence of a major draw. If a film has other big stars, it's possible they could "easter egg" them into the movie to give moviegoers something else to discuss at the water cooler.

Along with Kevin Spacey refusing any billing in Se7en in order to keep the killer's identity a surprise, Bill Murray had no screen credit in Tootsie and Owen Wilson was uncredited in A Night At The Museum. James Earl Jones similarly refused credit as the voice of Darth Vader in Star Wars, believing David Prowse brought the character to life, not him. Bruce Willis refused onscreen acknowledgement for Four Rooms; he was only doing the part for director and friend Quentin Tarrantino. And I'm almost positive Whoopi Goldberg goes uncredited as Guinan in all the Star Trek movies she's in, because she feels they are Star Trek movies and not Whoopi Goldberg movies.

First of all - the characters you mention in your question are most likely going to turn out to be Inhumans, as Marvel Studios, creators of the Marvel Cinematic Universe (of which Iron Man, Thor, Captain America, Hulk, The Avengers, Guardians of the Galaxy etc belong to) have no legal rights to use the term mutants in any capacity, as that term and related characters are owned by 20th Century Fox.

As for your main question - Yes, it's entirely possible, and has happened before. Kevin Spacey refused top billing in the film Se7en, directed by David Fincher and starring Brad Pitt and Morgan Freeman, resulting in his name missing from advertising and opening credits, ensuring that audiences were surprised to see him when he eventually does show up.

There have been plenty of examples given already. I think Kevin Spacey in Se7en is probably the most notable that I can think of, but the other example that popped into my head was Bill Murray in Zombieland.

There are a few ways to make this happen as well. For a very small cameo, it's possible to clear out the studio of all but the essential cast and crew for the actual filming of the scene, to ensure that as few people as possible actually see who the cameo is and what they're doing. This was the case for Nick Fury's cameo in Iron Man. Unfortunately, this is not totally reliable, as you can see in the article.

A quite common method is to shoot films under a codename. Again, this is not perfect, and for major films where casting has been announced or it's a sequel, it wouldn't be hard to walk by a filming location, see RDJ, Chrises Hemsworth and Evans, and Scarlett Johannson there and conclude that it must be a set for an Avengers movie.

In your specific case, as someone else has mentioned, it's just not legally possible, without a deal between Fox and Disney.

Scarlet Witch and Quicksilver are a special case, due to the fact that they've played such a prominent part in the Avengers for a long time, but are also some of the oldest mutants in the comics as well. The iterations of the characters in the different franchises show that they're not in the same universe though, as there's a much bigger age difference between the two in X-men than in the MCU (Wanda is a little kid in DOFP), and Quicksilver is probably around 18 in DOFP, and going to be not much older in AOU. I'd love to see X-men and Avengers together in a movie, but I don't see it realistically happening any time soon. What is more likely, though still doubtful, is that Spider-man may make an appearance in an MCU film at some point. Sony is having more trouble with Spider-man than Fox is with X-men.