He quickly runs. Not natural by itself, seems to be a forced "literary" style. But you could say: He quickly ran to the gate to welcome them.
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Leave off "quickly". He ran to the gate is enough, better.
If you mean to say he arrived quickly, and he ran to get there quickly then say:
He ran to the gate and arrived in ........ a few seconds, an hour, whatever. You are saying two things.
1. He ran - it is not necessary to say "quickly" - running is quick.
2. He arrived quickly - give it a time. Be more precise.