Travel from Shinjuku - Mt Fuji - Gotemba

Hello,
We will stay in Shinjuku during our trip to Japan in Dec. Would like to do a day trip to Mt Fuji and Gotemba. However not sure about the trip sequence( visit Mt Fuji first or Gotemba?) and how to get there? What time should we leave our hotel and what time is the last bus back to Shinjuku? Also would like to go to Yeti for skiing but not sure if this can be added to the same day trip? Would appreciate advise. Thank you.

Depends how close you want to go to Mount Fuji! Gotemba itself is located at the base of the mountain.

If you want to go to the Fuji Five Lake region, I recommend to go there directly from Shinjuku as explained on the following page:
http://www.japan-guide.com/e/e6905.html

And then take a bus from Kawaguchiko Station to Gotemba Station (one hour, 1-2 buses per hour), from where there is a shuttle bus to the outlet.

Then, take the bus back to Tokyo:
http://www.premiumoutlets.co.jp/en/gotemba/access/index.php

As for Yeti, I recommend to visit either Yeti or the Fuji Five Lake area. But packing all into a single day sounds too busy to me, especially if you don't have a car (access to Yeti by public transportation is bad):
http://www.yeti-resort.com/e/access.html