How long a cartridge lasts if you just vape it until it's empty depends on you. If you fill it yourself and keep it topped off, it should last anywhere from a few days to a week or more.
If you're ordering blank cartomizers, you should get at least 10 to start. By the time you get through 5 of them, you'll have a better idea of how many you need to order the next time. Remember, you can't mix flavors in a cartomizer. When you're starting out, you're going to be testing flavors. So unless you use an atomizer to test juice, you're going to be going through more cartomizers than you normally would.
If you're ordering pre-filled and don't have any extra juice to refill them, I'd get at least 10 or 15, maybe more. That should last an average smoker a week or two. It's really hard to estimate without knowing your smoking habits.
An average vaper goes through about 3-5ml a day, if that's any help.
And, I'd suggest manual batteries unless you have some overwhelming need to vape no-handed. The button becomes second nature in no time. It allows you much more control over your drag. An auto battery requires a "primer" puff to heat up, then cuts you off in some pre-determined time. It's also more susceptible to damage from leaking cartos. An auto battery can be drip resistant, but they're not drip proof.
Edit: O.k., a half pack on weekdays, pack on weekends. A cartridge is roughly, VERY roughly equal to about 6-10 cigarettes. Figure 8 cigs on average. That's about 11 cartridges per week if they're filled and you don't refill them.