How to perform function once per visitor after loading the site?

You are using PHP because it does not create and check the cookie with PHP. Then it only gives echo $tmp; if the cookie does not exist or has expired

<?php
$tmp='<script>
    Push.create("Olá Mundo!", {
        body: "Esta é uma mensagem nova",
        icon: "images/logo.jpg",
        timeout: 4000,
        onClick: function() {
            window.location="http://www.google.com.br";
            this.close();
        }
    });
</script>';


$biscoito=$_COOKIE[acessada];

if(empty($biscoito)){

   setcookie( "acessada", "sim", strtotime( '+1 days' ), '/' );  // 24 horas

    echo $tmp;

}
?>

cookies - php

empty -php

I would do it using the new localStorage of HTML 5, it's very simple and I believe that all modern browsers already have:

<script>
// escrevendo
localStorage.setItem("ja_acessou", "Sim");
</script>

<script>
// Lendo
if (localStorage.ja_acessou) console.log('já acessou!');
</script>

Example number of hits:

<script>
if (localStorage.qt_acessos) localStorage.qt_acessos += 1;
else localStorage.qt_acessos = 1;

</script>

I particularly like using cookies for this case, for example:

function getCookie(cname) {
    var name = cname + "=";
    var ca = document.cookie.split(';');
    for(var i=0; i<ca.length; i++) {
        var c = ca[i];
        while (c.charAt(0)==' ') c = c.substring(1);
        if (c.indexOf(name) == 0) {
            return c.substring(name.length, c.length);
        }
    }
    return "";
}

function showUmaVesPorDia() {
    var cooki = getCookie("msgCookie");
    if (cooki != "") {
        // ja existe o cookie msgCookie!
    } else {
           // não existe, vamos guardar!
           $("#mensagem").fadeIn('slow');       
           setCookie("msgCookie", 1, 1);
    }
}
$(document).ready(function(){
        showUmaVesPorDia() ;
});

Here's a working example link