Sadly, the monorails do not run all over the property. I'm assuming that you won't have a car, so you'll have to take a bus from ASMu. Depending on the MK hours, if the MK is still open the quickest route is probably the bus from ASMu to the MK. From the MK you can pick up the monorail which does go right into the lobby of the Contemporary (very cool, the kids will love it) . . . I think it's also a short walk from MK to the Cont.
Otherwise I think you have to take the Downtown Disney bus form ASMu and transfer at DD/Pleasure Island to a bus headed for the Contemporary (or, more complicated, DD to the ticket and transportation center to connect to the monorail).
Of course, depending on how tight your shedule is and how tired everyone is from a day of travel, you can always opt for a cab. They can call one for you at ASMu (and again on the way home from the Contemporary). We went to dinner at Boma on our first night (after spending the afternoon at Thyphoon Lagoon). Long story short, after waiting forever and ever and ever for the bus we were all very tired and cranky by the time we got to dinner. We opted for the short cab ride back to ASSp (well worth the $5-6) since using the bus system to go from resort to resort can be a nightmare.
Any way you do it, you'll have a fabulous time. Have a wonderful trip!!
p.s. If you do have a car that you haven't mentioned . . . drive! Resort to resort it's 100% worthwhile.