Not because he has broken into Celtic's first team while still a teenager. That's something the Bhoy who has jumped from Sunday League football to sitting on the bench against the likes of Barcelona and AC Milan has taken in his stride.
The heat Fisher is feeling comes from the fear he might be the man to make the mistake which brings Celtic's shut-out run to a shuddering halt.
The likeable 19-year-old will again be at right-back tonight as the Hoops attempt to stretch their unbeaten run in the SPFL to 23 games when Kilmarnock come to Parkhead.
He has proved a very able deputy while Mikael Lustig recovers from surgery and Adam Matthews is sidelined with a calf strain.
But the youngster is acutely aware that, the longer Celtic's shut-out and winning run continues, the more of a target their scalp becomes. His method of coping is to put it all out of his mind when he takes to the field and focus only on getting through the 90 minutes.
Asked if it brings extra pressure to be the new Bhoy in the settled and experienced back four, and not wanting to be one who makes the run-ending error, he replied: "It kind of does - but, I don't think about that. As a defender, I want to keep a clean sheet, anyway."
Fisher has already bought into the importance of "keeping it clean" every time he pulls on a first team jersey.
It's not something which is spoken about in the dressing room. It is simply understood by all involved.
"It is vital we keep the shut-out run going," said Fisher. "We are just doing our bit in helping Fraser, and he did his bit on Sunday against Hibs."
Fisher was a late developer, not coming to the attention of Celtic until he was leaving school as a 16-year-old.
A year earlier, he had been with Eldon Celtic in the Reading and District Sunday League when a side from the Jimmy Johnstone Academy travelled south to play them.
His performance encouraged the Academy to invite him to Glasgow to play for them in tournament organised by Rangers. Then a short spell with Farnborough in the Southern Conference was brought to an abrupt end when Celtic offered him terms in 2011.
Fisher admits he didn't know anything about Jinky - in whose memory the academy was founded - but has since done his research.
"I bought a DVD of him, and I have pictures of me with his statue when I came up on trial," he revealed.
Now it is Fisher who is driving down the right flank for the Hoops, aiming to add to the club's rich history.