Yes, you can get fittings in plastic to connect to cast iron, or imitation cast iron in plastic, or aluminium, cast is very expensive and you would need special brackets to carry the heavy weight, and is not recommended.

I would recommend to do a full replacement from the old cast iron to new plastic. You may have to replace the supporting timber fascias if they are rotten too.
You will need to dispose of the old cast iron pieces though, or leave out for the scrap collector.

Do not waste your time or money.
Get rid of it and put up plastic...
Brett Martin is a good reliable make

It would be better to replace the guttering with new pvc guttering,but it depends on what your budget is,if you can't go to that,I would suggest get a pvc gutter adapter and connecting to the cast,once fitted you can buy an fibreglass kit from Toolstation for £8 and apply that to the joint on the inside.

I would first look at the amount of Chanel and down pipe to replace all of the old. If it’s been up there since the place was built it’s about time for a change. And the cost of new nowadays it isn’t going to bankrupt the people and you know it’s all new and should last.

If the guttering is up vs 1930, it is best to speak to your neighbours. I seem to want to have all the guttering changed at one time. It will be more cost effective and a bit of job for the future. If not then. Yes, you can most certainly joining UPVC guttering to cast iron. There is many different types of adaptors. You can use your contractor should be well aware.