Yeah... This ain't an easy one to complete...
S-Ranking requires perfect play (no deaths/restarts), patience, perseverance, etc.
How I managed to S-Rank all the Town Missions I won't know.
I have 16 of the Achievements. The last 7 require me to S-Rank everything that's left and find all the Silver Medals.
I do know an easy S-Rank, though. In Silver's first Town Mission (The Apple Festival), if you commit suicide by jumping off the bridge nearby, you get to the results screen with more than 40,000 points and 0'00'016 on your time. Instant S-Rank.
I also know that to get an S-Rank as Silver in the Sonic vs. Silver fight, you need to practically defeat Sonic in less than 30 seconds. You just grab a bunch of stuff, jump, and mash X to throw the stuff at high speed.