When used after the 連用形（れんようけい） (conjunctive form) of a verb, either of the verbs 上（あ）げる ‘to raise’, or 上（あ）がる
‘to rise’ may be used. They will convey that something has been done ‘to the limit’. 上（あ）がる
will be used for actions that take place by themselves (intransitive), while 上（あ）げる
will be used for actions that have an instigator (transitive).
Although 上（あ）げる and 上（あ）がる
show that something has been done ‘to the limit’, the nuance is usually ‘to do (A) to the point of completion’, or ‘to do (A) through to the end’. Therefore, the ‘limit’ that is being highlighted is usually a natural ending, and not due to running out of some kind of resource. Let’s have a look at some examples.
Did you finish writing the paper?
If you use this rice cooker, your rice will finish cooking in 20 minutes.
Is it impossible to finish up writing this paper by tomorrow? I should give it my all.
He used sand to build up a small city.
上（あ）げる and 上（あ）がる
are most commonly used when some kind of effect or product is produced as a result of the (A) action. In these cases, the literal meaning of ‘to (A) up’ is very similar to ‘to finish up (A)’ in English.
His pieces are drawn up in 5 minutes.
It takes an old computer 10 minutes to finish up booting.
仕上（しあ）がる is a very common expression that is used for many things. Usually it conveys that something has been completed after some involved or time consuming process.
We will call you by your number once we finish wrapping it.
I wonder how that will be completed.