Is this the Sicilian Pistachio Cake from RHC? If so, it’s a very mild cake, and I’m not sure if strawberry mousseline would overpower it or not!
As to keeping it, if you mean just the “cake part” so you can frost it the next day, I do this all the time. I put each layer on a piece of waxed paper and then in a gallon size ziplock bag. The waxed paper makes it easy to lift out of the bag—it keeps it from sticking to the bottom of the bag. Then just put it in your fridge. I suck some of the air out with a straw, but gently—you don’t want to compress the cake. I never syrup.
If you want to make the whole cake and then serve the next day, just make sure that when you frost it that there’s no gaps between the frosting and your cake carrier (or whatever the platform is) so that the frosting “seals” the cake. Then you can just put the lid on and put it in the fridge. If you want to, you can wait until the frosting gets good and cold and firm, take it out, some plastic wrap on it, cover it again and put it back in, but I wouldn’t do it just for one or two days. Make sure to give it a good 4-6 hours to come to room temp for serving (unless you like cold cake, which I do).
Good luck! Report back!