As you are a runner and do have some fitness, all you really need to do is adapt to riding a bike.
So, I'm with VO2 on this one, you need to complete some rides that are greater in distance than are completed in the tour itself. Perhaps once or twice a week.
You may also wish to get used to riding a bike day after day so start by riding every other day and then try to string 2, 3, 4 and then 5 days of riding together. There is no need to make these sessions long at all, perhaps just over an hour each.
I guess that when you tour you will be riding quite slowly with only spells of harder riding (no sprinting or maximal effort). Therefore, most of your riding should be done at a fairly easy pace, similar or just above the pace that you are going to want to complete the tour at.
You may also want to put in a 'tempo' session once or twice a week to help prepare yourself for those efforts over hills or efforts to get out of the rain/to the next stop/pushing a friend. During the tempo session aim to go out for just over an hour, start of easily and build up to a level where your breathing is heavy, heart rate is rapid and you have to concentrate to maintain your pace. Hold this level until the hour is almost up, gradually warm down for 10 minutes. Don't do two of these sessions back to back.
If your tour will be hilly, choose hilly routes for 3 rides out of five and if the tour isn't hilly choose a hilly route for 1 out of five rides.
You'll need to be fresh for the tour itself, so reduce your hours in the saddle in the week before the tour so that you recover fully. Build gradualy during the training and make sure that you don't get two fatigued (it would be a shame to miss the tour due to illness).
When training and during the tour think about correct gear selection and pedal cadence as this will save you energy and help keep your legs in good shape for the whole week!
This is only guidence, if you want to get more serious check out some of the other strings that chat about endurance training and lifting lactate/anaerobic threshold.
You will need to think about your nutrition during the ride and between rides as well, you might get some help through the nutrition forum.