Each coffee has it's own special taste and distinctive flavours. Whether it is pre-ground Maxwell House or small-batch roasted single origin, there will be different flavour profiles and smells.
The "quality" of coffee is mostly determined by the bitterness and taste. Fresh, great quality coffee is often bright and has complex flavours that fill your palette. Diner coffee is usually bland and more bitter.
In high-quality beans, the enjoyment of the coffee is then more down to personal taste. That is, whether someone likes a bright, almost tea-like single origin from Ethiopia, or a deep-bodied Sumatran coffee with herbal notes.
This Kopi Luwak coffee will certainly have it's own distinctive taste, that's for sure. Will it taste better than cheap coffee like Maxwell house? Yes, probably. Will it be the best coffee you've ever had? Maybe not, and that is just fine. People drink this coffee because it is rare and sought after, not neccesarily because they are looking for the best cup.
As for brewing it, I would suggest trying it a number of ways, because each brewing method brings out different flavour profiles. French press will allow you to experience the body that is likely from a South-east asian coffee and will allow the natural oils onto your palette. Pour-over will take away some of that body but accentuate the herbal tones and bring out the top-end in terms of flavours. Espresso is always interesting, and the concentrated method could yield some surprising flavours.
If you have the means to try it, I would. It's a neat experience and if you are keen on coffee, tasting more coffee, no matter where its from, is always a good thing.
As for cleaning your equipment, just make sure to do a regular cleaning of your espresso machine/french press to remove any residue. This doesn't have to be intensive, just make sure there are no leftover grounds. For you grinder, I would disassemble as much as possible and wash and dry the parts. Grinders always retain a decent amount of grounds, and those mixing in with the expensive Kopi Luwat could change the flavour.
Here are three videos where people try this coffee:
Hope this helps!