You maybe can with your fleece. My understanding is that fleece pushes the wetness towards whatever is less wet. So, if you use a thick enough diaper, the fleece should work. But, if your diaper gets rather wet, then the fleece will pull it through. Some people like to use a layer of fleece inside the prefold to pull the wetness away from baby's skin.
To the OP-
I figure I'll need to change a newborn an average of every 2 hours, or about 12 changes a day. It's definitely possible to use more or less, though. I also like to go simple for all my kids in diapers. (I had two until recently, and ds still needs one at night. So, when this baby comes, I'll have 2 again, and 3 overnight).
So...I get Mommy's one size snap covers. It doesn't matter which cover I grab, it will fit any of my kids. Makes life much easier for me (and more importantly, dh.) He can never remember which cover is for which kid. The downside is they can't be reused quite as well as the others because they are fabric inside and out. The PUL is sandwiched in-between. I prefer it, though. If they are just wet, you can let them air out and then reuse them with a younger baby no problem.
That said, to wash every other day, I would recommend:
12-15 Mommy's touch covers
24-36 prefolds (minimum!)
Another thing to remember is that washing the prefolds takes longer and is a little more involved. The covers are usually good just going through once and line drying. So...if you need a cover, through one or two through with your regular laundry or whatever, and you're good. It's the prefolds that you really need to be sure to have plenty of.
And, of course, whatever wipes method you want to use. For the little ones, I just keep a basket of baby washclothes and get one or two wet with warm water from the tap.
Snappi's or pins if you want them. You'll have better luck reusing your covers if you snappi/pin your prefolds, but my babies seem to be too wide for the snappi's. (Pins aren't hard, though. Just stick them in a bar of soap first.