In terms of programmable logic, the DE1-SoC is about ~20x bigger, and has ~70x as much embedded memory. Practically any modern FPGA is bigger than the "Xilinx XC6216" cited by his papers, as was linked to you in the other instance of this question you asked.
That said, most modern FPGAs don't allow for the same fine-granularity of configuration, as compared to older FPGAs - the internal routing and block structures are more complex, and FPGA vendors want to protect their products and compel you to use their CAD tools.
In short, yes, the DE1-SoC will be able to contain any design from 12+ years ago. As for replicating the specific functions, you should do some more research to determine if the methods used are still feasible with modern chips and CAD tools.
Edit:
user1155120 elaborated on the features of the XC6216 (see link below) that were of value to Thompson.
Fast Configuration: A larger device will generally take longer to configure, as you have to send more configuration data. That said, I/O interfaces are faster than they were 15 years ago, so it depends on your definition of "fast".
Reconfiguration: Cyclone V chips (like the one in the DE1-SoC) do support partial reconfiguration, but the subscription version of the Quartus II software is required, in addition to a separate license to support PR. I don't believe it supports wildcard reconfiguration, though I could be mistaken.
Memory-Mapped Addressing: The DE1-SoC's internal data can be access through the USB Blaster interface. However, this requires using the SystemConsole on the host PC, so it's not a direct access.
