Assuming a 10 hour per day well sunlit panel, full output, you would only get 800 watt-hours per day. Now to be more realistic, 6 hrs. of charge, 80% output, 64w X 6hrs is 384 watt-hours.
Not much power there. You could run two 12v 11 watt CF lights for 17 hrs per day. This would consume the whole day's production and leave no room for charging batteries.
An inverter will also consume some power if you use one. How much it consumes will depend on the inverter.
I think I remember that batteries lose about 20% of the power input for charging.
If you charge all day and use no power, that puts you at just over 300 watt-hours per day charge into the batteries. That's 25 amp-hours at 12 volts. This doesn't take into account any wiring losses either.
Remote internet access is available by satellite. I think it is about $60 to $70 per month. It would require more power than you would get from this panel per day to operate it. Check out Starband, Earthlink satellite, AOL is also available by satellite. The buy-in is sort of high. Around $500, if I remember right, for the equipment.
Hope this helps. If I said anything wrong, someone please jump in and let me know.
Steve