6 days in japan

Is traveling to Osaka, Kyoto, Fuji and Tokyo possible in 6 days? And what trains is cheaper?

Doable but what would you do with one day in each location.

6 days = 1 location only. Pick 1.

Day 0 Arrive Tokyo
Day 1&2 Tokyo
Day 3 in morning take Shinkansen to Kyoto and look out of window to see Fuji (hopefully)
Day 3-6 Kyoto with daytrip to Osaka if you want
Day 6(or 7) depart from KIX or Itami airport.

Use Shinkansen 1 way ticket to Kyoto and local trains everywhere else.

It's possible, but I don't recommend it. Instead, consider:

If landing in Tokyo/Narita:
Tour Tokyo, Fuji, Kamakura
https://www.japan-guide.com/e/e2164.html
https://www.japan-guide.com/e/e6906.html
https://www.japan-guide.com/e/e2166.html

Or, if landing in Osaka:
Tour Osaka, Kyoto, Nara
https://www.japan-guide.com/e/e2157.html
https://www.japan-guide.com/e/e2158.html
https://www.japan-guide.com/e/e2165.html