It Depends. Probably not usefully.
As stated, the rule requires an invitation from an occupant before the vampire may enter a residence. (It says nothing about entering other spaces.) So we need to decide what these words mean in the context of the question.
An occupant of any given space is someone physically present in that space at that time. We all occupy Earth. Some of us occupy North America. And so on, down to the volume enclosed by your skin.
A residence is a space where specific people predictably sleep and keep personal items, distinguished from other spaces by social conventions. Here are a few examples from which to extrapolate:
- A free-standing house is one residence, even if three dozen undocumented construction workers sleep in shifts therein.
- An apartment building or row-house is one structure with multiple distinct residences.
- A cave may be a residence if someone makes a habit of sleeping in it, but not otherwise.
- A hotel room is a transient residence: it's a secure place to sleep and keep personal things unattended, but different people use it each night.
- The tent-city underneath the H1 freeway in Honolulu is not a residence, but it contains many residences, because the poor people who live there have divided it up into distinct areas each continually used by corresponding family groups.
- The common spaces in an apartment building are not a residence: nobody may live in the lobby or beside the swimming pool.
Residency: I would argue the prison works like an apartment complex: each cell is a distinct residence with a specific subgroup of prisoners who typically sleep there. In the common spaces, prisoners mingle and nobody sleeps.
Occupancy: Assuming you and I lived at opposite ends of an apartment building, it makes no sense for me to stand at my front door and invite the vampire to enter your living room, for I do not then occupy your residence.
Other Spaces: Presumably the vampire can enter workshops, stores, and the like without an invitation, so long as those spaces do not also constitute part of someone's residence. In particular, if there's so much as a cot or a blanket in the warden's office, it may be considered a distinct (if transient) residence rather than merely a work space.
Conclusion: The prisoner can invite the vampire to enter his own cell, but that is not relevant to any other part of the prison. The guard's dorm is off-limits in any case.