Yes get setup early but there is too early.
I would suggest at the speeds you will be doing in your piston twin the DOC of the ATIS won't be in an issue. Its not for me and we are rocking in at 250knts.
10-15 mins before you expect to be handed over to the approach controller get the wx don't worry if its going to change in the next 5 mins. Brief the approach get the radios set up then a quick check to see what the changes in the wx are and then wait for the handover. If you think that you going to be speaking to the approach controller just as it changes don't worry about it. They will either spot that its just changed and tell you whats new or you can have a quick listen just after speaking to them.
The getting the wx off fisbangwallop is real life ops not for IR tests. Have the volmet frequencys but I suspect you won't have time to wait through the whole list of airports. I can't remember if I had to listen to it or not.