Windows 98 runs on MS-DOS version 7, using the FAT32 filesystem which does not enable files to reveal as much information about themselves as NTFS is capable of.
Beyond the long filename, the 8.3 DOS filename, the Attributes for Hidden, System, Readonly, Archive and Directory, and the file size, there are three Date/timestamps - Modified, Created and Accessed - which is the information you require.
However, Accessed is a date-only field, with all times represented by 00:00 as far as I can see.
Third party file management tools were available for Windows 98 by the shedload on computer magazine cover disks in the late 1990's, just before the internet took over as the major means of distributing small software shareware and freeware items.
Now there are very few utilities that will run on anything less than Windows 2000 or XP. Someone will now prove me wrong about this. :) Still, your system is networked, so there's still some hope that a utility will work across the network.
As far as utilities I know that work on Windows 7 and 98 (and everything inbetween), Karen's Directory Printer http://www.karenware.com/powertools/ptdirprn.asp
will certainly give the Filename/Accessed_date (plus any other selected available data) in the form of a text file.