The difference between Alister and Amumu is Alister's tanking abilities aren't as item intensive. Amumu can do great damage if you focus on his AP aspect. If you want to make him a hybrid tank/caster, try going Rod of Ages into Abyssal Scepter and Sorcerers Shoes for more AP/spell penetration. While I've seen people put Nashor's tooth on him, it doesn't really fit into his role as tank/initiator, making Frozen heart a better item for CDR, but a little later in the build.
Honestly, with the way I play Amumu, I make great use of Lichbane in conjunction with a constantly recharging tantrum. Rod, Abyssal, Lichbane, and Mercury Treads give you good HP, a ****-ton of MR, and fantastic damage.
I've seen effective Amumus go Chalice and Mercury Treads into dual heart of golds and then dual sunfire capes. The first 4 items are dirt cheap and make you an all around solid tank in the midgame, giving you HP, armor, and MR. Late game when you have despair maxed out and a sunfire cape or two, you'll do great AoE dps. Its a cookie-cutter tank build which works on quite a few different tanks.