Any time I've been at WDW when Tortuga has actually been open it has only been for very limited hours around lunch time. I've only gotten to eat there once in May of 2014 and I got there shortly before they closed for the day. The only way to know if it is open is by checking the operating hours schedule when you get to the park for the day.
I would doubt it would be open last weekend in January. It's never been open anytime I've been there, but I go in "off" seasons. I think it's only open during busy times.
Tortuga Tavern is open only during busy times so it's unlikely it will be open at the end of January and when it is open the hours are fairly short - basically just for lunch from 11:00 to 3:00. I was surprised to see it open during mid October last year, but by the last week of the month it was closed.
End of January isn't a busy time, but when it has been open, it has only been for limited hours at lunch, not all day. Check the times guide when you arrive at MK.
We've only eaten there twice, both times in the summer. Honestly, the food wasn't that good, probably my least favorite in the MK, so you won't be missing that much anyway. Hope you have a wonderful trip!
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