For someone with mobility issues (and lord knows I have them myself at the moment) best advice for a pleasant and active day out, and as good a chance as nearly anywhere of seeing Fuji San, is to go on the Hakone circuit. You get there by train, then transfer to another train (that goes through some marvellous scenery), then hop to another train that's pulled by cables up some very steep mountain sides (you can have a little wander around the town of Gora if you're up to it - some great hot springs there) Then it's an easy transfer to some aerial cable cars past big boiling valley (which is where you might first see Fuji), a change to some more modern aerial cable cars (going down this time) and a transfer to one of the open decked ships that ferry across lake Ashinoko - where you can also see Fuji (if she isn't hiding that day). From there it's a bus to the start of the circuit.

If it's just a day trip, get your concierge to tell you if there's a probability of Fuji being seen that day (if you're at the Park Hyatt, you should be able to see for yourself) if not, choose another time to try.

Hardly any walking required and lots of ground gets covered easily.

You'll need to get the Hakone Free Pass - there are two kinds:
http://www.odakyu.jp/english/freepass/hakone_01.html
and weekday pass: http://www.odakyu.jp/english/freepass/hakone_02.html

More info here: http://wikitravel.org/en/Hakone

This photo was taken by dsf and shows lake ashinoko on a not completely clear day: http://flickr.com/photos/dsf/266534019/ It was taken with his phone, and he says Mt Fuji looked a lot sharper than his picture suggests (it's actually about the furthest point from where you'd see Fuji on the circuit)


Don't forget to get a price from www.tokyoairporter.com and compare it to what you've been offered elsewhere (if you try and acuire a guide with car from the Japanese Tourist Board be prepared for some astoundingly high quotes - an elderly man who had trouble walking wanted to see Nikko with a guide & approached JTB - I sent him on the bullet train to Kyoto and got Mr Doi to escort him around the city in his car instead - turned out to be much, much cheaper!)