In Catholicism, is a priest allowed to administer the "last rites" to himself (assuming no one else is there to do it for him)?
Can a priest administer last rites to himself?
The short answer is no.
The Roman Catholic sacrament of anointing of the sick or extreme unction is performed on a seriously ill person for spiritual and physical strength, or when a person is close to death as preparation for heaven. The priest anoints the sick person with oil and prays over him. When combined with confession and the Eucharist, it is called “Last Rites.” - What is extreme unction / last rites?
A dying priest may give himself the Eucharist, but he can not absolve himself of his own sins. That takes another priest. Extreme unction is performed by a priest on a sick or dying person.
It is highly doubtful that a priest is permitted to anoint himself sacramentally. This sacrament is performed by a priest on someone who is extremely ill, not on oneself. Like confession, it takes two people to make the sacrament valid: a priest confessor and a penitent.
It can also be requested for those who are unconscious or who have lost the use of reason if they would have asked for it were they in control of their faculties. - What is extreme unction / last rites?
I personally know a bishop, who as a priest anointed unconscious many people at an accident scene conditionally by starting each anointing thus: "If you are Catholic..."