IMHO coding an Arduino sketch to interface to the board will be *FAR* easier than trying to hardware hack an old DOS era EPROM programmer to do so. However if your C programming skills are non-existent and you've never dabbled in any other procedural programming language, YMMV.
Parallel EPROMs have separate /CE, /OE and Vpp pins and possibly a PGM pin. The RAM used on the ICOM memory board, and thus the board as a whole, doesn't. It has three control pins, one of which: WP, is the write protect for the lower 1/4 of the memory (via the 74LS32), which in a programming jig is disabled by tying it high, and can therefore be ignored. The other two are AD10, which drives the RAM's /CS pin, and /WR which drives the RAM's /WE pin.
A RAM read cycle only requires /CS to go (or be) low. However for a write cycle, /CS and /WE must be sequenced low and released in the correct order to avoid a data bus conflict as there is no /OE pin. Its unlikely that any EPROM algorithm has compatible signal sequencing - you'll have to scope it and see, without the RAM board, but with 10K pullup resistors on the data bus. As some EPROM algorithms are interactive - reading back each location to check if its programmed then calculating and applying an overprogramming pulse (or pulses), that may not be good enough and you may have to test with a blank EPROM to see the full signal sequence.
You must deal with the PGM voltage - at the very least it will need limiting to logic levels before input into the logic you will need to add to derive /CS and /WR from the EPROM signals. Also many of the EPROM programmer's algorithms will boost the supply voltage when programming, but over 5.5V is undesirable for the ICOM board.
Hopefully you wont blow up the RAM board . . . .
A universal programmer is a different and far easier beast to make an adaptor board for, as it will already have algorithms for programming NVRAM modules, which keep all signals within the 5V logic range, and with luck, one of its RAM algorithms will have the right sequencing and timings for a uPD444.