Can two male neutered cats stay together?

Cats are not small dogs and can't be treated the same way; where dogs are generally social with one another, adaptable, and can easily be moved between locations for petsitting and quickly introduced to one another, cats are more solitary and do not handle disruptions to their environment well.

To your titular question, yes, two neutered males can live together, and frequently do. However, you cannot just throw them into the same room and expect it to work out as it does with dogs; they will fight, possibly injuring each other, necessitating vet visits for one or both.

You don't have to decline the pet sitting because of this, however, you should ideally conduct the pet-sitting at the owner's home if at all possible, not bring the cat into yours. A healthy cat can be checked in on a couple of times a day and will be fine on its own the rest of the time. If the cat has health problems that necessitate a sitter staying with it, you should decline in this case unless you have someone at home to care for your cat; health problems should unquestionably stop any thoughts of trying to bring the cat into your home or introduce the two to each other, as the stress could make those problems much worse. For more information on how to cat sit, see my answer to First time cat sitting - what should I do?, as well as the answer to Pet sitting best practice for information for your friend to prepare.

If you insist on bringing the cat into your home, Recommendations for introducing adult cats to newly adopted adult cats and Using the room door as a means of introducing cats to each other slowly, as well as other questions related to introducing cats to one another, can help, though your best bet will likely be to just keep the two isolated from one another for the duration of the pet sitting.