Even if it takes the whole day, I'll finish typing that up.
Literal
Even-if-たとえ one-day entire take-ても that typing [object-を] finish-up.
仕上げる ('to finish up, complete, put the final touches on') is a compound verb: 仕 ('do') + 上げる ('raise up, finish'). The suffix 上げる adds the 'bring to completion' nuance to many verbs: 書き上げる (write to completion), 作り上げる (make to completion).