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If you're having mana regen problems, I'd recommend you get a Doran's Ring at level 1 or work on Chalice of Harmony instead of Clarity. You should get ghost instead, because it's more useful mid to late-game. It's important to conserve your mana so that you only use it when a teammate is ganking your lane.
When I lane, I prefer to go AP, because tank Amumu is just so good at ganking when he's jungling. I don't know if AP Amumu can jungle. Whichever build you're going to use really depends on your team composition.
You should also get Despair before Bandage Toss. It does a lot more DPS than you would expect, especially on tanks.
Your build has a few problems. First of all, you're getting Warmog's armor mid to late-game. This is generally a bad idea, because you cannot help your team in teamfights or you'll get ganked while farming. If you're getting Warmog's Armor on any champion, you may want to consider rushing it. Another problem is that you get Randuin's Omen. This slows down your enemy attack speed and that greatly hurts your E (This applies to Frozen Heart). Your E's unique effect is what gives Amumu his damage. A few items that I'd recommend getting instead are Guardian Angel, Abyssal Scepter, Soul Shroud, or Shurelya's Reverie (Rod of Ages, Zhonya's Hourglass, Rabadon's Deathcap, and Sheen are good items if you want to go AP).