If I have a 4 day MVV City Tour Card & Tagekarte ticket, can I get to the Neuschwanstein Castle using that ticket?
No - Neuschwanstein is far outside the MVV area. The cheapest solution is a Bayern Ticket (25€/one, 31/two).
Yes, even though Neuschwanstein is often seen as a day trip from Munich, it's several hours each way from Munich - not "close" at all.
The best you can do, if you have a Gesamtnetz Tageskarte, and
1) you want to use a Bayern-Ticket,
2) you want to start before 9 AM on a workday,
3) you will leave on a train that is still in the MVV area (Gebiet) at 9 AM.
then you can use your city-card to get to the stop after 9 AM and the Bayern-Ticket thereafter.