Watts are a measure of power, not capacity. A rating of 650 Watts indicates the maximum power that can be delivered, but says nothing about how long it can be delivered for, which is a separate measure.
There is also a measure Volt-Amps, which is related to the Wattage, and says how much current can be delivered at a particular voltage. With a DC supply (such as a standard car or consumer battery) they are identical, although because the voltage is fixed battery ratings are usually in Amp-hours or milli-Amp-hours; but in an AC supply the Watts consumed by a device are the product of the voltage and current only if the two are in phase - if not the power consumed is less. A power supply is still limited on the current it can supply, even if there is little power consumed, hence the VA rating.
The less power taken from a battery, the longer it will last, though the relationship is far from simple. So a power supply that can deliver 100W for an hour will be able to provide 50W for longer, but whether it is more or less than two hours will depend very much on its design and the batteries it uses.
Your modem draws 10 Watts (not Watts per hour), so your UPS needs to be able to deliver this power. This is a very low demand and most will be able to meet it. Many manufacturers provide graphs of the length of time against power drawn, and you need to look at these carefully. They may offer a modular solution to allow the UPS to be configured to deliver whatever continuity you require. In your case I would specify double the expected maximum power cut, eg specify 48 hours to provide a day's back-up. This is because batteries become less efficient with age, and the UPS may not deliver its initial rating after five years.
