Problem/question regarding my newely bought Sony D6C

What about their manual to unsolder something and then speed could be tuned?

 

The adjustment described is for W&F stability, not for tape speed.
They say desolder the bridge (which disconnects the PLL, so no quartz lock), then set speed to 3090Hz (using a 3000Hz test tape).
Then re-solder the bridge.

No matter what speed you set up with bridge disconnected, when you reconnect it speed will be back to what it was before.

If you think there is a problem and speed does indeed seem too low audibly, look elsewhere:
- clean the capstan ring with IPA to make sure there are no traces of oil or grease;
- etch the capstan with ferric chloride so it has the best grip possible;
- replace the pinch roller with a new one if you haven't done so already;

Seen a couple of situations on DD walkmans where a polished capstan (despite it wasn't visible by naked eye) caused the speed to be low.

 

As per you advice when I have gotten this walkman I vhanged all rubber parts including pinch roller. About capstan, can I use 2000 geit or higher sandpaper? will this work?

 

Fine grit sandpaper will polish the capstan even more. Ferric chloride needs to be used as we want to roughen the surface to have more grip.

Procedure is simple: you put the walkman in rewind, soak a cotton swab in the solution and rub it against the capstan in the area where tape rides.
Be careful not to apply too much solution, as it will drip into the capstan bushing and you really don't want to etch that.
After clean with IPA both the capstan and roller and see if the situation has improved.