Run time for a solar christmas lights

The question can not be answered with the information supplied. You need to give the current draw of the bulb and the amp/hour (AH) rating of the battery. Then all you do is figure out how long the bulb will last on that battery.

For instance, if the bulb draws 1A and it is a 10AH battery you would be able to run the bulb for 10 hours. If the bulb was a 500mA bulb with the same battery it would last for 20 hours.

For the temperature low you are indicating the AH rating of the battery would be much less, unless you keep it in a heated room (normal room temperature). To know how much the battery would be derated at -20C you would have to contact the manufacturer of the battery and find out if they have a graph that shows the battery capacity at different temperatures.

Of course there is always the sneaky way to do it. Take a bulb, connect it to a fully charged battery (that is already in -20C temperatures and has sat there enough to bring it's temperature down to near -20C.) Now start a stopwatch and watch it until the bulb goes out. There's your answer.

You don't usually want to run a battery down completely. Most battery-based solar power systems limit the battery minimum state of charge (SOC) to something around 40% to maintain the life of the batteries. If you are just powering a christmas display, that may not be so critical.

Are you also asking about the battery charging capablity of the system? If so you need to know the size (in Watts) of the solar module and the amount of sunshine (W/m^2) in your area. Do a search fro "PV Watts" and use that service to find the amount of sun you get in your particular area.