You want to provide a shared Zettelstore that can be used from your various devices. Installing Zettelstore as a Linux service is not that hard.
Grab the appropriate executable and copy it into the appropriate directory:
# sudo mv zettelstore /usr/local/bin/zettelstore
Create a group named zettelstore:
# sudo groupadd --system zettelstore
Create a system user of that group, named zettelstore, with a home folder:
# sudo useradd --system --gid zettelstore \
    --create-home --home-dir /var/lib/zettelstore \
    --shell /usr/sbin/nologin \
    --comment "Zettelstore server" \
    zettelstore
Create a systemd service file and store it into /etc/systemd/system/zettelstore.service:
[Unit]
Description=Zettelstore
After=network.target

[Service]
Type=simple
User=zettelstore
Group=zettelstore
ExecStart=/usr/local/bin/zettelstore run -d /var/lib/zettelstore
WorkingDirectory=/var/lib/zettelstore

[Install]
WantedBy=multi-user.target
Double-check everything. Now you can enable and start the zettelstore as a service:
# sudo systemctl daemon-reload
# sudo systemctl enable zettelstore
# sudo systemctl start zettelstore
Use the commands systemctl and journalctl to manage the service, e.g.:
# sudo systemctl status zettelstore  # verify that it is running
# sudo journalctl -u zettelstore     # obtain the output of the running zettelstore