ASICs by their very nature can attack any known problem.
For the problem of randomly accessing gigabytes of data, ASICs can't beat commodity DRAM. That's why there's so much research into memory-hard PoWs like Momentum, Dagger, and Cuckoo Cycle.
You're absolutely right. ASICs can include ram internally or use external ram modules just like a computer. General purpose processors are generally cheaper and more effective for memory intensive applications.