Guess I'm having trouble with the word "lift" in calling it a boat lift. My mental picture of a boat lift, picks the entire boat out of the water. The link in your post is one used to pull a boat on to a trailer, or a dolly which will pull it on a track up a ramp of sorts.
So in that case the Armstrong draws 6 amps at 120VAC and has a 3 minute duty cycle. So you can run it for 3 minutes then it needs to rest for 20 minutes. Not to many folks pay any attention to duty cycle, but this is the main reason things burn up over time. Has a 4000 lb line pull
You can get a 12V DC 4000 lb winch for under $200. It uses a planetary gear set (128, 2:1) to gain the power to pull at 4K lb
http://www.amazon.com/4000lb-Electric-Winch-Corded-Remote/dp/B00C1QEQJC
But back to your question, you might be able to find a 12V motor which could be used, but you would need to do some gear reduction to achieve the same power. With the 12V winch you could use a solar charger and battery, but this is another issue. The standard solar changer does not put out much. It would not be able to recharge the battery in a day. You would need to get a larger solar panel to produce enough power to recharge.
All depends on how often it would be used
