I would hope if these two teams are in the quarter finals, they are of a reasonable standard (for their age) - so I wouldn't be expecting foul throws particularly. The best advice I would give is ref it as you would any other game. Some players of that age like to ape their heroes: they will question decisions, they will look at you if they think they've been fouled with their arms spread wide and a look of astonishment on their face if you are saying no foul, they will want you to stop the game to tie laces if they want to force a break in play, they will be looking for penalties, there will be shirt pulling and so the list goes on.
As
@haywain said, you may well get grief from the sidelines - decisions will be questioned (very loudly), and you may well receive abuse. Here I would say remember the age of the players. If you don't like the tone or the language being used, nip it in the bud early.
Does the youth league allow CARs to do more than in and out of play (the youth league I ref on does)? If so, brief everybody at the same time as to what you want from them, and remember to say that if you have a different view on something you'll indicate to them to lower the flag and carry on. Remind them that when they are running they line they are not allowed to "coach" (and my approach to that is I don't want to hear a peep out of them).
Also, no coaches or spectators behind the goal lines - they should be on one side of the pitch behind a respect barrier, with the coaches/subs the other side of the pitch.
I would reiterate though, ref it as you do any other game - it may be a hotly contested game as its a quarter final. Most of all, enjoy!
NB/ U13s are 35 minutes each way, 10 minutes each way added time if a draw at the end of normal time, before going to penalties.