corkgirl, we will be making a couple of port stops in Ireland this summer. I do have some questions, and will post them in our Ports of Call section.
Regarding your questions, January is low season in the Caribbean. January is the coldest month of the year in NYC, but it's often tolerable. According to weather.com, the average high is 2C and the average low is -5C. That compares to 11C and -4C in Shannon, according to the same source.
I've never been to the Caribbean in the summer, but this will be peak period, as Cruise Fanatic noted. I have been in NYC in the summer (many, in fact, as I grew up there), and it can get hot and humid. If early May is an option, weather is great in the Caribbean, and NY is beautiful in the spring.