Yes, it is open in February. From December 13 through 25, it is closed because of its annual maintenance.

Tell me when you arrive at Shimizu port. If my schedule is free, I will guide you without a travel agent, which is much more convenient for both of us.

I did an arranged bus tour from the ship which is the best way to go. A taxi would probably be very expensive. I don't recall seeing many taxis at the cable car station so a taxi would need to wait for you and the trip on the cable car is quick, but then you would spend time at the shrine before heading back on the cable car. There is a public bus that stops at the ropeway but it's not clear how you would pick it up from the cruise terminal. Check their website (which is listed here) for detailed information on ways to get to the ropeway. Good luck!