Battery Charger

So let me first start off by explaining my project then explain what I need help figuring out.

My arduino will be running off of a lipo battery but will only be powered on for about 1 minute each day. Using the board found here ( https://www.pololu.com/product/751 ) I will be able to turn the arduino on via a push button then have the arduino turn itself off by sending a signal to this board. When the arduino is not on this board uses extremely low power. However once the arduino is done doing what it needs to do there will be a power supply charging the lipo battery back to full charge.

What I need help with is finding a board that will allow me to charge the lipo battery but also allows me to use it’s energy when needed. However whenever the battery is not charging I need the charging board to use very little power. Basically I need to make this battery last without any issues.

I am looking at the board found here ( https://www.sparkfun.com/products/12711 ) and might be using that. Let me know of your thoughts.

I would not use a LiPo for something that runs only 1 minute per day. With proper power savings techniques, a couple of AA batteries could last for years. See http://www.gammon.com.au/power You may also be able to use a solar cell and supercap, as described in this interesting article: http://www.gammon.com.au/forum/?id=12821

I like this idea about the super capacitor. My question is, what happens if the capacitor is fully charged but is still taking energy from the solar panel?

Also how long would it last if I used this super cap https://www.sparkfun.com/products/746 with a 5v step-up