What software I need to buy to work with Cyclon III dev. board?

IPS-EMBEDDED contains the IP components typically used in FPGA based embedded processor systems: NIOS II, DDRx controller, Ethernet stack. As other Altera IP licenses, they are not time limited, but are not guaranteed to work with any future Quartus version.

The Quartus subscription version contains only the memory controllers from IPS-EMBEDDED suite, there have been also special offers of the subscription version that contained a NIOS II license. All-in-all, IPS-EMBEDDED suite is an interesting offer, if you only need the respective IP components, I think. Without a license, you're restricted to time-limited evaluation runs of your design.

In my understanding, the suite can be used with free Quartus Web version without restrictions. There may be an implicite restriction, if Altera decides to require a new license for a new Quartus version someday and you won't be able to receive a new web license for the old Quartus version...