Use /usr/bin/systemd-socket-activate to mimic systemd's socket activation protocol, or implement a custom tool that does the same. (This specific tool only supports delay-starting, i.e. when the first client connects, not immediately, as testing that was its original purpose.)

An equivalent might be:

#define _GNU_SOURCE
#include <err.h>
#include <stdlib.h>
#include <stdio.h>
#include <sys/socket.h>
#include <sys/un.h>

int main(int argc, char *argv) {
    char *basedir, *mpddir, *addr, *pidstr;
    struct sockaddr_un sa = { .sun_family = AF_UNIX };
    int r, sock;

    if (argc < 2)
        errx(1, "mpd executable and args not provided");

    basedir = getenv("XDG_RUNTIME_DIR");
    if (!basedir || !basedir[0])
        errx(1, "XDG_RUNTIME_DIR not set");

    r = asprintf(&mpddir, "%s/mpd", basedir);
    if (r < 0)
        exit(3);

    r = mkdir(mpddir, 0700);
    if (r < 0 && errno != EEXIST)
        err(1, "mkdir(%s)", mpddir);

    r = snprintf(sa.sun_path, sizeof sa.sun_path, "%s/mpd/socket", basedir);
    if (r < 0)
        exit(3);
    if (r >= 108)
        errx(1, "socket address too long");

    sock = socket(AF_UNIX, SOCK_STREAM, 0);
    if (r < 0)
        err(1, "socket");

    r = bind(sock, (struct sockaddr *) &sa, sizeof sa);
    if (r < 0)
        err(1, "bind");

    if (sock != 3) {
        r = dup2(sock, 3);
        if (r < 0)
            err(1, "dup");
        close(sock);
        sock = 3;
    }

    r = asprintf(&pidstr, "%d", getpid());
    if (r < 0)
        exit(3);

    setenv("LISTEN_FDS", "1");
    setenv("LISTEN_PID", pidstr);

    execvp(argv[1], argv+1);
}

Use /usr/bin/systemd-socket-activate to mimic systemd's socket activation protocol, or implement a custom tool that does the same. (This specific tool only supports delay-starting, i.e. when the first client connects, not immediately, as testing that was its original purpose. Shouldn't be a problem – if you want music to play on boot, you can just follow it up with an mpc call to immediately poke the socket and cause mpd to be activated.)

In some distributions, systemd-socket-activate might be in /usr/lib(exec) instead.

If you don't want to have the systemd tools installed either, then an equivalent might be:

#define _GNU_SOURCE
#include <err.h>
#include <stdlib.h>
#include <stdio.h>
#include <sys/socket.h>
#include <sys/un.h>

#define LISTEN_FDS_START 3

int main(int argc, char *argv) {
    char *basedir, *mpddir, *addr, *pidstr;
    struct sockaddr_un sa = { .sun_family = AF_UNIX };
    int r, sock; 

    /* The intended usage is "./mytool /usr/bin/mpd --systemd --blah" */
    if (argc < 2)
        errx(1, "mpd executable and args not provided");

    basedir = getenv("XDG_RUNTIME_DIR");
    if (!basedir || !basedir[0])
        errx(1, "XDG_RUNTIME_DIR not set");

    r = asprintf(&mpddir, "%s/mpd", basedir);
    if (r < 0)
        exit(3);

    r = mkdir(mpddir, 0700);
    if (r < 0 && errno != EEXIST)
        err(1, "mkdir(%s) failed", mpddir);

    r = snprintf(sa.sun_path, sizeof sa.sun_path, "%s/mpd/socket", basedir);
    if (r < 0)
        abort();
    if (r >= 108)
        errx(1, "socket address too long");

    sock = socket(AF_UNIX, SOCK_STREAM, 0);
    if (r < 0)
        err(1, "socket(AF_UNIX) failed");

    r = bind(sock, (struct sockaddr *) &sa, sizeof sa);
    if (r < 0)
        err(1, "bind(%s/mpd/socket) failed", basedir);

    /* Systemd socket activation protocol defines that FDs start from 3.
     * Normally this little program will only have 0-2 when it starts, as
     * usual, so the socket() call will also issue a FD starting from 3
     * but it's better to make sure and move the socket to 3 if needed.
     */
    if (sock != LISTEN_FDS_START) {
        r = dup2(sock, LISTEN_FDS_START);
        if (r < 0)
            err(1, "dup failed");
        close(sock);
        sock = LISTEN_FDS_START;
    }

    r = asprintf(&pidstr, "%d", getpid());
    if (r < 0)
        abort();

    setenv("LISTEN_FDS", "1");
    setenv("LISTEN_PID", pidstr);

    execvp(argv[1], argv+1);
}