There is more than one way how to use the two power jacks on RB450 (the obvious one being - if you use another type of enclosure, and don't want a wire in the front).

This creative example shows a battery attached to one of the jacks. The jacks are connected. As this jack receives less voltage, so it's charging the battery, and board is powered from the powered jack. If that jack would be removed, the battery would kick in.

Batteries that can be used are 12V from 1-5Ah. PSU switching adapters are recommended 13-13.8V so that they can charge these batteries. The battery in the picture is lead-acid jelly type. From this 2.2Ah battery, this device will work around 2 hours at full load.

I'm about to buy my first RB450G, will such scheme work over POE-powered RB ? As it will be very nice to have such powersupply in places where hard to find 220v.

No, 13V is too low for POE supplying. It is mostly for indoor usage.

Batteries that can be used are 12V from 1-5Ah. PSU switching adapters are recommended 13-13.8V so that they can charge these batteries. The battery in the picture is lead-acid jelly type. From this 2.2Ah battery, this device will work around 2 hours at full load.

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Can the power supply be higher than 13.8V like 15V?

No, 13V is too low for POE supplying. It is mostly for indoor usage.

I think it is not to hard to calculate input voltage, so on device end we will receive 13v from POE.
Other question is will be those ports powered in case of POE?

This decision is bad:
1. Insufficient charge the battery.
2. Deep discharge batteries. Small life of the battery.
3. No monitoring.