Javascript ES6, 905 octets
Ouf .. L'horloge de 12 heures a tué ma solution autrement courte ( ahum ). 
905 octets pour imprimer 898 caractères d'ascii art yay
_=>(a=(d=Date)(c=new d(),z=O=>Math.abs(new d(a[3]+O)-c)).split` `)[1]=='Apr'&a[2]==20|~a[4].search(/(04|16):20/)?'0,0,2ca2o,0,10e,0,0,1eu9z4,0,22w,1ulajuo,0,bklxc,2zgg,i,120gdfk,8lc,1s1s,15,j0jf9c,7em8,tac,10,190zcw0,tm8w,ksk,hra0ia,mihog0,3aels,wy,iv8e15,b98u80,cvs1s,575,9fm71g,190zcw0,bg7vgg,1t4,148d4lu,121pblx,b9ea68,2b36,iv8hn5,9zleso,15kytz4,98cp,4fwpwk,zir9ud,1h9u3ln,11c6vb4,84kda8,kbuo05,b98u84,14possy,k9su6i,a26hwk,zvft34,kavrpc,1tn1j4,1r3jis,kamgao,a5bhq8,2b2q,beyt6x,j0y810,muhiww,9xl,a5b7s8,14npcwi,1bkfw1s,4ys,14lipw4,kalr6k,0,1aq,kalqzg,0,48m,2j6hqf,zik0zk,0,33mcjk,0,5'.split`,`.map(a=>('0'.repeat(32)+parseInt(a,36).toString(2)).substr(-32)).join``.match(/.{1,3}/g).map(a=>` .:\`|;'
`[parseInt(a,2)]).join``:~~((T=Math.min(z('/4/20'),z(` ${a[1]} ${++a[2]} 0:4:20`),z(' 4:20'),z(' 0:4:20'),z(' 16:20'),z(' 0:16:20'))/1000)/60)+` minutes and ${~~(T%60)} seconds left until your next hit.`
Code non golfé
Remarque: Le code non golfé ne s'exécutera pas
function() {
    // (all vars here are actually global)
    var d = Date; // short reference to Date object
    var dateString = Date() // "Tue Apr 12 2016 20:13:00 GMT+0200 (W. Europe Daylight Time)"
    .split` `; // ["Tue", "Apr", "12", ...]
    var currentDate = new Date();
    dateDiff = function(toAppend) {
        return Math.abs( // no dates in the past
            new Date(dateString[3] + toAppend) // new Date('2016' + something)
            - currentDate // substract the current date. 
                          // Thanks JS for the datetime implementation
        );
    }
    if ((dateString[1] == 'Apr' & dateString[2] == 20) |
        ~dateString[4].search(/(04|16):20/)) { // if the time doesn't contain 04:20 or 
                                               // 16:20, this will return -1. ~-1 is 0 
                                               // and thus falsy
        alert(printAscii());
    } else {
        alert(printTimeLeft());
    }
}
function printAscii() {
    var magicString = '0,0,2ca2o,0,10e,0,0,1eu9z4,0,22w,1ulajuo,0,bklxc,2zgg,i,120gdfk,8lc,1s1s,15,j0jf9c,7em8,tac,10,190zcw0,tm8w,ksk,hra0ia,mihog0,3aels,wy,iv8e15,b98u80,cvs1s,575,9fm71g,190zcw0,bg7vgg,1t4,148d4lu,121pblx,b9ea68,2b36,iv8hn5,9zleso,15kytz4,98cp,4fwpwk,zir9ud,1h9u3ln,11c6vb4,84kda8,kbuo05,b98u84,14possy,k9su6i,a26hwk,zvft34,kavrpc,1tn1j4,1r3jis,kamgao,a5bhq8,2b2q,beyt6x,j0y810,muhiww,9xl,a5b7s8,14npcwi,1bkfw1s,4ys,14lipw4,kalr6k,0,1aq,kalqzg,0,48m,2j6hqf,zik0zk,0,33mcjk,0,5' // ;)
    var splitted = magicString.split`,`;
    var binarySequence = splitted.map(function(piece) {
        return ('0'.repeat(32) + // 000000000... this will be padded
            parseInt(a,36) // parseInt('190zcw0',36) -> 000000002722627584
            .toString(2) // -> 10100010010010000000000000000000
        ).substr(-32) // left pad binary sequence
    }).join``; // create one big binary sequence from several smaller
    var groupsOfThree = binarySequence.match(/.{1,3}/g);
    return groupsOfThree.map(function(three) {
        return ` .:\`|;'\n`[parseInt(a,2)]; // magic.. turn 010 into : etc
    }).join``; // create them blaze
}
function printTimeLeft() {
    // this is way longer than it should be.. probably
    var minimumTimeDifference = Math.min(
        dateDiff('/4/20'), // -> Difference between now and 2016/4/20 00:00:00
        dateDiff(' 4:20'), // 2016 04:20:00.. Sadly the 2016 is needed (i think)
        dateDiff(' 0:4:20'), // 2016 00:04:20
        dateDiff(' 16:20'), // Damn 12 hour clock :(
        dateDiff(' 0:16:20'),
        dateDiff(` ${dateString[1]} // 2016 Apr 
                   ${dateString[2]+1} // 2016 Apr 13 (12+1)
                   0:4:20` // Check the first occurrence of 4 minutes and 20 seconds 
                           // the next day
    )
    var timeInSeconds = minimumTimeDifference / 1000;
    return Math.floor(timeInSeconds) + // ~~ works like Math.floor
        ` minutes and ${Math.floor(timeInSeconds%60)} seconds left until your next hit.`;
}
Essayez!
f=
_=>(a=(d=Date)(c=new d(),z=O=>Math.abs(new d(a[3]+O)-c)).split` `)[1]=='Apr'&a[2]==20|~a[4].search(/(04|16):20/)?'0,0,2ca2o,0,10e,0,0,1eu9z4,0,22w,1ulajuo,0,bklxc,2zgg,i,120gdfk,8lc,1s1s,15,j0jf9c,7em8,tac,10,190zcw0,tm8w,ksk,hra0ia,mihog0,3aels,wy,iv8e15,b98u80,cvs1s,575,9fm71g,190zcw0,bg7vgg,1t4,148d4lu,121pblx,b9ea68,2b36,iv8hn5,9zleso,15kytz4,98cp,4fwpwk,zir9ud,1h9u3ln,11c6vb4,84kda8,kbuo05,b98u84,14possy,k9su6i,a26hwk,zvft34,kavrpc,1tn1j4,1r3jis,kamgao,a5bhq8,2b2q,beyt6x,j0y810,muhiww,9xl,a5b7s8,14npcwi,1bkfw1s,4ys,14lipw4,kalr6k,0,1aq,kalqzg,0,48m,2j6hqf,zik0zk,0,33mcjk,0,5'.split`,`.map(a=>('0'.repeat(32)+parseInt(a,36).toString(2)).substr(-32)).join``.match(/.{1,3}/g).map(a=>` .:\`|;'
`[parseInt(a,2)]).join``:~~((T=Math.min(z('/4/20'),z(` ${a[1]} ${++a[2]} 0:4:20`),z(' 4:20'),z(' 0:4:20'),z(' 16:20'),z(' 0:16:20'))/1000)/60)+` minutes and ${~~(T%60)} seconds left until your next hit.`
alert(f())
 
 
Dans l'extrait de code suivant, vous pouvez définir la date pour tester l'art ascii
day=prompt('Day?', 12);
month=prompt('Month?', 'Apr');
f=
_=>(a=(d=Date)(c=new d(),z=O=>Math.abs(new d(a[3]+O)-c)).split` `)[1]==month&a[2]==day|~a[4].search(/(04|16):20/)?'0,0,2ca2o,0,10e,0,0,1eu9z4,0,22w,1ulajuo,0,bklxc,2zgg,i,120gdfk,8lc,1s1s,15,j0jf9c,7em8,tac,10,190zcw0,tm8w,ksk,hra0ia,mihog0,3aels,wy,iv8e15,b98u80,cvs1s,575,9fm71g,190zcw0,bg7vgg,1t4,148d4lu,121pblx,b9ea68,2b36,iv8hn5,9zleso,15kytz4,98cp,4fwpwk,zir9ud,1h9u3ln,11c6vb4,84kda8,kbuo05,b98u84,14possy,k9su6i,a26hwk,zvft34,kavrpc,1tn1j4,1r3jis,kamgao,a5bhq8,2b2q,beyt6x,j0y810,muhiww,9xl,a5b7s8,14npcwi,1bkfw1s,4ys,14lipw4,kalr6k,0,1aq,kalqzg,0,48m,2j6hqf,zik0zk,0,33mcjk,0,5'.split`,`.map(a=>('0'.repeat(32)+parseInt(a,36).toString(2)).substr(-32)).join``.match(/.{1,3}/g).map(a=>` .:\`|;'
`[parseInt(a,2)]).join``:~~((T=Math.min(z('/4/20'),z(` ${a[1]} ${++a[2]} 0:4:20`),z(' 4:20'),z(' 0:4:20'),z(' 16:20'),z(' 0:16:20'))/1000)/60)+` minutes and ${~~(T%60)} seconds left until your next hit.`
alert(f())