The recommended approach would be to have udev start a systemd service, which itself depends on your device.
The service file should look something similar to the following:
my.service - to be placed in /etc/systemd/system
[Unit]
Description=<DESCRIPTION HERE>
BindsTo=<DEVICE UNIT HERE>.device
After=<DEVICE UNIT HERE>.device
[Service]
ExecStart=<CALL TO SCRIPT HERE>
Note: to get a list of the available device units use
list-units --all --full | grep ".device"
And the udev rule should be something like the following:
90-my.rules - to be placed in /etc/udev/rules.d
KERNEL=="tty*", ATTRS{serial}=="<DEVICE SERIAL HERE>", TAG+="systemd", ENV{SYSTEMD_WANTS}="my.service"
Note: to get a list of the attributes of your specific device, including its serial number, use
udevadm info -a -n /dev/tty*
This question, though fairly different, might also be of interest.