Gotemba is a little over an hour from Tokyo on the Tomei Expressway, and Mishima or Shizuoka about an hour by shinkansen. That makes it an easy extension of a Haneda itinerary or a Tokyo trip.
The Fuji-area courses divide between Gotemba in Shizuoka and the Fuji Five Lakes in Yamanashi, while Izu offers onsen-resort layouts. The two Fuji axes point in different directions, so decide which one you are visiting before planning.
The Fuji view is clearest in winter, though higher courses run cold and snow can limit operation. In summer cloud often hides the mountain, so the view is not something to count on.
Comparing a Japanese tee time on the headline green fee alone tends to produce a different number at the counter. Confirm whether cart, caddie or self play, lunch, locker and the local golf tax are included, then check rental clubs or shoes, minimum player counts, two-player supplements, payment method and whether overseas visitors can book. Clubs may also specify check-in times, dress codes and bathing etiquette, so submit player names accurately.
With two nights, pick one axis. Drive from Tokyo, play Gotemba, stay at an Izu or Gotemba onsen, play a Fuji-view course on day two, then finish with the outlet mall or sightseeing.