It's rather an unusual use of "eilen" in this case. "Eilen, etwas zu tun", much like "sich beeilen, etwas zu tun", means "to do something in a hurry".
You can hurry up in order to do something:
Anne has to hurry up in order to catch her train.
Or you can do something in a hurry:
Anne has to go to the train station in a hurry in order to catch her train.
The distinction between the literal sense of "The priest should hurry while administering the last rites in time" and "The priest should hurry [while going there] in order to administer the last rites in time" is hard to make. The last rites traditionally have to be done with before the person dies, so "noch rechtzeitig" actually refers to the goal of being done by that time. This could be the reason why "eilen, zu ..." is used here instead of "eilen, um zu ...". In my opinion, both would possible here, but the author for some reason opted for the former.
To answer the question from the headline: no, in modern German, you generally cannot just omit the "um". You can use a "zu" infinitive clause with some verbs, but then it generally has a different meaning than an "um zu" clause. "Um zu" is a finale Konjunktion, which means it always states a goal, "zu" doesn't.
Other comments and answers suggest that Stefan Zweig uses an archaic German here where it was in fact possible to omit "um". For example in the apostolic creed:
"Von dort wird er kommen, zu richten die Lebenden und die Toten."
That's possible, but it would have to have to be interpreted as archaic/artificial on purpose then - this was out of use when "Schachnovelle" was written.