Hi FBB
I spent a few weeks up around Alice Springs around Christmas time, it was a great summer to be up there as there was plenty of rain around and a lot of the creeks and rivers were flowing.
Like NNW said above Emily and Jesse Gap are close enough to town that you could access on a bike, some walkers have been known to walk the ridge line between the two gaps. To the west Simpson Gap is doable by bike easily enough- theres even a bike path out there and there are some longish walks out there like the Woodland Walk and he walk out to Bond Gap.
Its a hard place to get around without your own transport but if you have time on your side then hitching is a reasonable option up there, the novelty of a bushwalker seems to enhance your chances of getting a ride.
If you want to check out some walks around Alice click on Northern Territory on my blog
http://goinferalonedayatatime.blogspot.com.au some of the walks are recent and some are from years ago, the map at the bottom of each post will show you where the walk is in relation to Alice Springs.
It should be a great trip.
Cheers Kevin