It sounds like a tendon strain at the Insertion
point of the biceps to me (origin is up in the shoulder joint).Arm wrestling places lateral strain on a hinge joint so the structural systems tend to struggle.You probably have a case of 'golfers elbow' (the opposite to tennis elbow) with the pain on the 'inside' of the bicep/forearm Training through the pain is not the sharpest thing you could have done but cest la vie, its done now.
You need to ice the area for 10-20 mins on 5-10 mins off, as continualy as possible for the next 48 hours. From here, start alternating heat and cold, with gentle stretching and mobility straight after the heat cycle. Do this for another 48-72 hours. Do no
upper body work whilst doing this (yep, take the week off buddy), and I would recommend wearing tube-grip support during day to day activity during the this recovery stage.
When the pain subsides return to light upper body work, (compound movements only-no direct arm work) stopping before failure/fatigue.If you remain pain free, gradually return to natural intensity and volume of training, being mindfull to warm your elbow joints up thoroughly before each workout.
If you insist on arm wrestling, learn to controll your oponents wrist by articulation of their thumb in your grip, you wont need so much effort then