I have had the exact same experience once when blood was drawn.
My veins are very visible so it has never been a problem before but this one nurse didn't stick the needle in on a wide angle to just enter the vein but had a larger angle and stuck it deep into my arm. Right away I said it was very painful and felt bad. He said that it's probably just touching a nerve and shouldn't be a problem.
So went home and had a tingling sensation there for the rest of the day and just figured it would pass.
The next day I went to the gym and did some weights. Every time I bent my arm with weights I felt a shooting pain through my arm and couldn't work out anymore. That's how painful it was.
I went to the hospital and had them check for nerve damage, had an x-ray for veins and another specifically for nerves. There was no visible damage luckily.
Long story short, it lasted 6 months and nothing could be done about it. A year later I still felt it with heavy weights.
It's annoying but if will pass eventually.