I’m not sure if I express it correctly. Below are variants. Can you help to choose?

Don't say 'scores'. Probably points, depending on the type of competition.
Say 'his nearest . . . '.
'Competition' is more idiomatic than 'contest'.

I'd say something like this.

After the first day of the competition, Mikhail was leading / was the leader, 5 points ahead of his nearest rival.

Here are a few comments.

After the first day of competition Mikhail led, outstripping the nearest rival by 5 scores.
'Outstripping' is not a common word in speech.

After the first day of competition Mikhail got on top, keeping the nearest athlete (rival) 5 scores behind.
During the first day, he got on top. After the first day, he was on top.

After the first day of contest Mikhail was a leader, edging out the nearest athlete (rival) by 5 scores.
'Edging out' sounds a bit like the competition is finished and Mikhail won.

Best wishes, Clive