Can a pulled muscle in neck effect the nerve that help extends the fingers?

Yes.: The nerves that control the sensation and strength/movement originate from the cervical spine i.e. neck. So injuries to this area can cause nerve irritation. Be sure to check with your doctor if your symptoms persist.

Radial nerve: The terminal branch of the posterior cord of the brachial plexus if compressed would cause wrist drop. Injury in the posterior cord itself would also impair the ability to abduct the arm past the first 15 degrees. There may also be weakness with abduction of the thumb and supination of the forearm.