We want a solar pump that will lift water from a base through the middle of an urn that is 4ft tall, where it wll flow down to the base again. The lift-hose would be about 1/2 in or 3/4 in diamter. What specs do we need to look for?

Also, are there any kind of battery charging systems available whereby the solar cell would charge the battery for use at night, for 2-3 hours at a time? I would mount the panel in a place beside the fence that gets 8 hours sun a day, then bury the cable to the pump 8 feet away in the courtyard.

How big a panel does one typically need for such an application? Any links to DIY or assembly kits and/or advice would be welcome!

For your pump specs, you need the head (height) you need to pump to and the flow rate at that head.
The head is the vertical distance from the surface of the water in the basin that the pump is in to the top of the urn -- sounds like around 4 ft in your case.
To get an idea of the flow rate you need, I would take a garden hose, place it in the urn as you would the pump output hose, and adjust the garden hose flow until you get the flow you want. Then measure the garden hose flow rate by timing how long it takes to fill something like a gallon jug -- if it takes 30 seconds to fill the gallon jug, then you want a pump that will pump about 2 gallons/minute (gpm) with a 4 ft head.

As far as sizing the PV panel, its best to pick the pump first, and then get a battery and PV panel that match the demand of the pump. For example, if you get a pump that draws 1.2 amp at 12 volts, thats (1.2)(12) or 14 watts.
Maybe something like a 15 or 20 watt panel.

You can add a battery for night use. You will need a small charge controller between the PV panel and the battery to keep the panel from overcharging the battery.
You might look at some of the small systems they sell for RV battery charging as one possibility.

Gary