Around the same time, the clerk Natsue Arimura submitted her resignation to Gokigami, saying this month would be her last.
Literal
Same time [target-に], clerk Natsue Arimura [topic-は] this month [object-を] limit [to-に] resign [genitive-の] intent [object-を] Gokigami [target-に] apply-submitted.
Narrative novel style. 今月を限りに ('with this month as the limit') is a fixed formal expression: 'with X as the end/cutoff.' 旨 ('purport, gist, intent,' read むね) is a formal noun for the content of a statement, often used in business/legal writing: ~旨を伝える ('convey the content that...'). 申し入れる ('to submit, propose') is a formal verb for putting forward a proposal or declaration. Typical of crime/drama narration.