Pokedoc's tool is your best bet: http://projectpokemon.org/forums/showthread.php?12405-A-tool-to-backup-HG-SS-saves
If your Action Replay GBA Duo can run homebrew (it most likely can't), then that's the easiest way
If the MicroSD in your ARDSi is 2GB or less, you don't need anything else but Wi-Fi (unsecured or WEP; WPA won't work). You just need to DLDI patch Pokedoc's tool with the Games 'n Music patch. If you don't know how to do that, you can use DS Organize. Run DSOrganize-gnm.nds, then use that to run the tool I linked to above. DS Organize is a bit unstable when writing to the MicroSD, so I don't recommend doing anything but use it to DLDI patch things.
If you don't have compatible Wi-Fi, then you could consider a USB adapter, like the NDS Backup Tool Plus, SMS4, or R4i Backup Adapter. Keep in mind those devices don't play nicely with 64 bit Windows, and will require some bypassing (see the cons in the OP). I have had little luck using them recently, so either Windows 8 changed something making it no longer work, or I'm doing something wrong.
If you have soldering experience and are brave, you can look at this guide, but I don't recommend it.
If none of the above apply to you or you don't have enough hardware, you have a few options:
-Get unsecured or WEP Wi-Fi. For security purposes, if you do this, you should only leave it running when you need it for your saves.
-Get an EZ Flash 3in1.
-Get any slot 2 flashcart that can run NDS homebrew.
-Get a usb adapter, like the NDS Backup Adapter Plus, SMS4, or R4i Backup Adapter. (If you're on a 64 bit Windows OS, you'll have to do some extra steps, which may or may not work.)