In whole song I'm counting notes: 1 2 3 4. Because I can see that the shortest note is 1/4. But there is an exception in first measure, last line. There is 1/8 note. When we have 1/8 notes then we are counting: 1 and 2 and 3 and 4 and. The question is: Should I count in such way for all song because I can see at least one 1/8 note? Should I count 1 and 2 and 3 and only in measure with 1/8 notes? How to do it properly? I try to play this song with metronome.
Because this song has simple rhythms and there are only two eighth notes in the entire song there is no need to count the “and” for the entire song but there is nothing wrong with counting all the “ands” if it makes you more comfortable and helps you play the song correctly. Your idea of just counting the “and” for bar 13 is a good one. The rhythm in that bar will then be:
ONE and TWO and three and FOUR AND
The numbers and “ands” in capital letters are where the notes fall.
In this song, counting 1 2 3 4 in each bar will work just fine. You will reach a stage when you won't even need to count that, you'll just feel it.
On the bar in question, most would count 1 2 3 4&, obviously the 1 2 3 4 will remain at the same pulse as they did in all the other bars. Don't see the need for all the & counts, but if it helps, there's no damage done.
EDIT: when you use the metronome, you would benefit from making it click on all eight counts - 1&2&3&4&, or at very least 1 2 3 4. That way, you play exactly with the clicks.
You make the shortest note value one and then go up. Quaver would be one count, crotchets two, minims four, semi-breves eight. Always make the shortest note value equal to one and then make the divisions make sense upwards. Start slowly and then start counting faster as you get the hang of it.
Playing a presto piece, andante, but in time, is less of a fault, than playing a piece's tempo correctly, but getting the rhythms wrong.
This is just to understand how to play the rhythm, it is not an explanation out of a theory point-of-view