This a moderate to strenuous experience regardless of your itinerary. You should come with a good level of fitness, as those who are “mountain-fit” when they arrive in Antarctica will enjoy it far more.
For climbing and mountaineering you should mimic the movements of hiking up and downhill with a backpack, scrambling over rocky ground while wearing boots and traversing steeper slopes. Doing this in a gym does work but sometimes being creative and taking a regular hike round your local park taking in all the uneven ground will be far more realistic and of more benefit.
Hiking with a pack is of use but be very careful when training with weight and don’t overdo it! You are better off developing your overall cardiac fitness than your strength to carry a heavy pack. Running, stair climbing and bicycling also build up your leg strength and aerobic fitness. Climbing walls are a great way to not only improve your upper body fitness but they also boost your confidence, coordination and technique.