I wanted to play as a fairy, and thought that a good way to introduce my character to the party would be with a scroll that casts conjure fey without the cast time or duration, but the spell brings CR into the equation.
You summon a fey creature of challenge rating 6 or lower
This is a level 3 campaign if that helps.
This is a cool backstory and character introduction idea that has no effect of game balance or mechanics, so just ignore game balance and mechanics for the sake of introducing your character in a cool way. You’re already making a concession on how the spell’s duration works, so you should fit it into the narrative however you like, ignoring other rules details for the sake of writing a cool and fun story. So for things like this, just work with your DM to work this into the narrative.
As for the actual question asked here, the CR of a player character is going to vary wildly based on their build. Two PCs of the same level could have vastly different CRs depending on how well optimized they are for combat, so there is no clean formula for converting character level to challenge rating.
It's great to be thinking about how to introduce your character. Discuss it with your DM. The DM could very easily come up with a plausible scenario like you are describing, without worrying about things like CR. The party could just find an enchanted figurine, or a hidden shrine, or any of a number of other things.
So, talk to your DM.
A level 3 PC would probably be, at most, a CR 3 creature, and is definitely not more than CR 6.
According to the Monster Manual, a CR 3 encounter is one that a typical level 3 party will find to be "a worthy challenge, but not a deadly one". If the individuals are of equal power, and it's four against one, that matches the "not deadly" part of the description.
