I am unsure if I am on the right track, but so far I understand this much:
The key to this riddle is only for you
The key is the phrase "only for you"
Below are instructions, above this the clue
Telling us the important phrase to start with
First strike the one near the head of a year
Remove a 'y' from the phrase, results in "onl for you"
Then remove he who begins a cheer
This is where I am stuck, logically I would remove 'c' since 'c' begins a "cheer" but there is no 'c' in the phrase
Next take away the end of a tunnel
End of a tunnel is 'l'. Removing that leaves us with "on for you"
Then let us go to solve this puzzle
Not sure if this line is important or just to rhyme with tunnel
What you first took you must now take again
The first character we removed was 'y', removing it again leaves "on for ou"
With three you are left, but fret not dear friend
We're left with 3 words
Fattest to front and thinnest to rear
Again, I am unsure at this line, possibly sort according to number of letters? But "on" and "ou" contain 2 letters each.
Add in two "eyes" and all becomes clear
Adding 'i' to 2 positions should spell out a number.
This is my reasoning so far, trying to figure out the rest.
Edit, interesting observation by M Oehm
"Striking" the one near the head of a year could refer to the 'l' in "only" as it is 'near' the 'y' (head of a year). Turning it into "onty". However, that makes the "tunnel" hint harder as there is no more 'l' to remove.
About the hint:
Seeing as how he focuses on the title and his literal word meaning so much, imagine the answer turns out to be "a number". The answer is in the title! Fits the hint.