There is no way you want to have a failed bicycle on the side of the road. Given the time you are taking for this experience, it makes sense to ensure your bike NOT an inconvenience along the way. Take your bike, even if it is an embarrasing nothing special bike like mine, to a reputable bike shop and have them go through it from stem to cassette.
A full tune up should handle all of this, but just in case, in descending order:
New tires and tubes. Road tires. Bring the old ones along, they will make nice spares.
New brake pads. Bring the old ones along, they will make nice spares.
Have the chain checked. Replace and lube as necessary. If you replace it, 10 links as spares.
Rear derailleur adjustment.
Shifts should be clean and dependable. Mechanism should be quiet.
Brake and shifter cable inspection and/or replacement.
Extra handlebar padding as necessary. New, padded gloves.
Headset and bottom bracket should be free and snug.
Have the wheels trued as necessary. A warped wheel will create a hot spot on a long down hill.
Check out pedals, clips and shoes.
Ride the bike a lot after it has been checked over. Ensure it works exactly right.
Ditch any extra weight.
TWO bottle cages OR a two bladder hydration system. Water + sports drink.
Have WHITE reflector mounted forward, RED reflector visible aft. This is a REQUIREMENT!
If you have a choice of bikes and a big budget, you want a mountain bike geartrain, road bike tires and road bike steering stability ("trail") and mountain bike handlebars with disk brakes. Having said that, an ordinary road bike will do, as will an ordinary mountain bike with road tires. It's just that some of the climbs will be tougher on a roadbike and descending on a mountain bike will get scary if your bike is twitchy at all. Anything heavy will take a toll on the uphills.
You may consider swapping seats.
Lots of saddle time. Get used to your choice long before the ride.
Since the only hammer options are UP hills, aerodynamic profile is less important than leverage and weight. Bicycle weight is more important to a 120lb rider than it is to a 220lb rider.
Speeding aerodynamically downhill is a bad idea. Hybrid bars are fine. NO aerobars.
Triple chainrings up front are a really good idea. If you can't do triple, talk to your bike shop about a smaller ring and a longer derailleur lower arm.
Whatever you choose, choose it ahead of time so you get used to the configuration well ahead of your ride.
An Awesome Week on a Bicycle!