Here is a short document for IBM 3584/TS3500 firmware upgrade.
Get the firmware package
Upgrade firmware via Library web interface
Service -- > Firmware update
Click Launch Firmware update Wizard
A new window will pop up and let you 'Select Firmware Type', at here you can choose to upgrade 'Library firmware' or a list of type of drives you want to do fw update.
In case you choose a type of Drive, then select either a single drive, or all drives
Browse the fw file, then there you go.
Traditionally, you would like to use the ways above to perform drive firmware update, but sometime they may fail due to slot connectivity, or other reasons. If that happens, then you can take a try to Use IBM Tape Diagnostics tool(ITDT) to update drive firmware
Or use tapeutil, the previous tool to ITDT.