La = a Lei ~ to You sir/ma'am
prego = I beg/request/ask
di arrivare presto = to arrive early
~ I request that You arrive early.
~ Please come early (sir/ma'am) . . . perhaps with a slight bow.
If you are not close friends grown up middle class Italians are usually very polite and formal when talking to each other. You may say salve but never ciao to your teacher, your manager, customer, elderly etcetera.
Here is a helpful article verbs and the prepositions that follow them. Pregare is not one of the verbs that is followed by an infinitive. It is followed by di + infinitive
I've generally only seen "arrivare" mean "to come" in more metaphorical contexts. For example, a court case coming to trial would "arriva in tribunale." Or "la morte arriva per tutto." Since this sentence is referring to a more literal arrival, "arrive" is probably the best translation. If the Italian used "venire," then "come" would work in English.