You can grow raspberries in containers, many do this because they spread otherwise and can be invasive in a garden.

Watering needs will depend on the quality of your soil (how much water it retains), how much sunlight it gets, and how much mulch you use. Your plant will let you know when it needs water as the leaves will lose rigidity, start to wilt, etc. As a rule anything in a container needs more water than something in the ground.

It will survive with cloud or shade, it will not thrive, and you will have less fruit production. More sun == more fruit.

Mulch when the current mulch decays away, you should tend to want to keep 1-2 inches of loose mulch on top of the soil. If you use a good quality potting mix it will have fertilizer added to it already, a little more wouldn't hurt, but it really wouldn't need a lot until next year.

Every day just make sure the leaves are water stressed.

Raspberries often do not need to be pruned, at all, but you may need to stake it at some point as they tend to flop over when they get bigger and have fruit on them. If you want to prune it, do it immediately after it stops fruiting, and cut it down to about 3 or 4 feet high. Doing it later can result in less berries next year. Raspberries will live for two years, then they will die, but the roots will put up new replacement plants automatically as it grows, eventually you will have too many plants in your container and will need to take some out to give away/throw away/put into new containers or it'll get overcrowded.

The larger container you can put it in the better. If you can get a big half barrel size container, it'll be quite happy in there.