First of all, I would like to point out that I don't like passthroughs because I have seen a few of them blow up unattended. If you don't have it yet, I recommend against getting one. I'd recommend you get yourself a pair of eGo C Twist replacement batteries instead. Then you'll always be able to charge one while the other is in use. This isn't a true backup though because if you lose one you're going to be hurting. It is not wise to charge your batteries overnight, so that means that in order to be safe you will need to charge it while you are awake. The average battery charges in 2-3 hours depending on the mah rating (1000mah is closer to 3). The reason why passthroughs can be dangerous is because people leave them plugged in unattended and as long as it's fully charged and plugged in, not only is keeping it 100% charge bad for the battery it puts its safety circuit to the test. If it fails, the thing will overcharge and vent (i.e. fire). There are several stories of this happening on ECF IIRC.
Second of all, you will want a Vivi Nova Mini
tank. Normally I recommend the E1-V over the Vivi Nova Mini because it looks better, doesn't require a thread collar/sleeve to cover the ego threads and look good, it screws metal onto metal (unlike the VN Mini which is metal onto plastic = crossthread and it's game over), and the E1-V has better flavor production BUT they are out of stock. Eric (from DiscountVapers (the company that sells them)) has no clue what's going on with the shipment from China. It's in limbo right now. The Vivi Nova Mini isn't bad, it's just not as good as the E1-V IMO. What you won't want to do is get a full size Vivi Nova tank on a tiny eGo battery. It will be extremely top heavy and will look like a mini mace/lollipop. I can't recommend that. I can't recommend the eGo Vision Clearomizer (Stardust) either because in my experience 8 out of 8 of them cracked and quickly became useless.
About battery life: 650mah (milliAmp Hours) is supposed to last approximately 6.5 hours of frequent use, but this isn't really precise because everyone uses their battery differently as we all have different habits. The resistance atomizer that you use, which in turn produces a certain wattage will have great effect on the battery life. The higher the resistance (and the crappier the vape on this low voltage device) the longer the battery will last, and vice versa. Let me try to break it down for you in a completely not understandable way.
I'm not an expert in this regard but, if you wanted to go into the theoretical, 650mah means that it can put out .65 of an amp for 1 hour. If you use an Ohm's Law Calculator you will see that at 3.6v if you put a 2.0ohm atomizer on there it will run at 1.8amps. 650mA/3=1.95 amps (close enough). That effectively cuts the 6.5 hours into a third of an hour-ish. In a rough estimate that means that the run time of that battery will be ~20+ minutes in operation if you consider that 1200 seconds of 10 seconds puffs = 120 total puffs. If you take 10 second puffs, one every 5 minutes, then that's 12 an hour and you could go 10 hours like that, theoretically. Someone please correct me if I am wrong about this.
I hope that helps. If you have any further questions please refer to the Vaper's Handbook (link is in my signature) and you will find most of your questions answered there already.