Another option to get close to Mt Fuji is to visit the city of Lake Kawaguchiko. Very easy to access from Tokyo or even on the way back from Osaka, Kyoto directions. There are Five Lakes that surround Mt Fuji, and by far the most easy to get to is Lake Kawagiuchi by Train or Express Bus. Lake Kawaguchi is also a Hor Springs resort, and you can stay in many of the high quality Japanese Tradtional Inns (Ryokan). Located on the lake's eastern side, Lake Kawaguchi has amazing views of Mt Fuji which is only approx. 30 minutes away. If you are lucky you can even stay in a Hotel room that has a private view of Mt Fuji.
Obviously Mt Fuji views can not be guaranteed every day of the Year. There are some months that are more likely to have Mt Fuji visible than others. Generaly mornings are better for the Mt Fuji views, and trust us, it is worth getting up that little bit earlier and wake up to be greeting by the 3000+ mountain.
Late afternoons are also great for views and if it is a fine day you may also be able to see Mt Fuji even if there are clouds covering the summit area.
There are many other attractions available around Lake Kawaguchi, such as hot spring baths, various museums, and the Fuji Q amusement park. On and around the lake itself, there are boat tours and a ropeway leading onto a small mountain next to the lake. Plus Lake Kawaguchi has uncountable numbers of Nature and Hiking tracks around the lake.