The OS should be no problem as it is a 386 hardware.
You'll probably need 4MB of RAM (likely higher) at an absolute minimum to run Linux, and likely 16MB to run any distribution or kernel with a decent software selection since 2000 or so. If you can't upgrade the RAM you are stuck.
Some brief searches seem to suggest this has an 8088 with 512KB or 768KB or RAM, though. Modern Linux won't run on that at all. (You may want to keep an eye on ELKS, the Tandy's NEC CPU is mentioned in the boot/setup.S file.)
I did get Linux booted on an old 1995-era "Winbook" laptop via floppy, I believe I used muLinux.
rs-232c connector
The way to "convert" serial to a network connection is PPP. You would need to set up a PPP client on your laptop, and have a pppd running on another Linux/Windows host that can route your ppp connection to your outgoing Internet connection.
You can probably still use it as an ssh terminal somehow if you install SSHDOS on it.
If anything, put an RS-232 adapter on your Linux system, configure your inittab to spawn a getty on ttyS0 or ttyUSB0 and use a DOS terminal program to access your system.