If you don't really need the WiFi capabilities you'd be better off using a 'standard' Pi Zero, as opposed to the Pi Zero W. According to these figures on raspi.tv the Zero uses around 20mA less on average - not a huge saving, but significant if you're running on batteries. Whatever solution you use, the battery's likely to be a hell of a lot bigger than the one you've linked to in your question - Pis are not particularly easy on the juice!
Using the raspi.tv figures linked above, if you run a Pi Zero W 24/7 for the duration (the simplest solution) you're looking at something like:
230 mA x 0.17 hours (10 minutes total taking photos per day)
120 mA x 23.83 hours
= 39.1 mAh + 2859.6 mAh
= 2898.7 mAh
2898.7 mAh x 7 = 20290.9 mAh
At the time of writing, Anker manufacture a 26800mAh power supply that should be large enough to run the whole thing for a little over a week. Your mileage with this may vary depending on temperature and longer term battery fatigue. This approach is as easy as it gets - just charge the battery, plug the Pi in, come back at the end of the week and take it away. No additional effort required.
For a more complicated approach you'd be looking to wake up and shut down the Pi at fixed intervals. There's no way to do this without using an additional microcontroller - something somewhere has to be on all the time to maintain a clock. There are some suggestions in this thread already - from those I would recommend looking at the Witty Pi add-on board (designed specifically for the job), or just using an external timer circuit and a relay to flip power on and off when you need it. I'd imagine that would get you down to something like 10 minutes of Pi uptime per hour (couple of minutes to boot, couple to shoot, couple to shut down, couple for safety margin), plus the 24/7 power consumption of your external circuit (if it doesn't have its own battery). The Witty Pi claims 1 mA of current use in normal operation, which would get you towards something along the lines of:
24 events x 230mA x 0.03 hours (2 minutes-ish) = 165.6 mAh
24 events x 120mA x 0.13 hours (8 minutes-ish) = 374.4 mAh
1 mAh x 24 hours = 24 mAh
165.6 + 374.4 + 24 = 564 mAh
564 mAh x 7 days = 3948 mAh
That's obviously a heck of a lot less power used, but there are a lot more potential points of failure there.
Whatever approach you finish up using, I would strongly recommend sticking with the Pi's micro-USB power input over connecting a battery directly to the 5V rail. The micro-USB connection offers a lot more electrical protection, reducing the chances that your Pi will burn if anything weird happens. It may cost a little more for a pre-built charger pack but you significantly reduce the likelihood that you'll destroy your Pi during normal operation.