You asked whether we can omit the preposition if there is no object after the verb 'arrive'.
Yes, you're correct. We DO use a preposition after arrive as in “arrive at 7 p.m.” or “arrive in the U.S.”, but if there is no time or place following, we don't need a preposition, as in:
“When did you arrive?”
“The train arrived late.”
“My friend arrives (on) Monday.”