Yes Prozac Side Effects do go away eventually and usually after a couple of weeks of taking the Prozac the side effects such as headaches, nausea and trouble falling asleep should go away.
If the side effects after taking Prozac don't go away within a few weeks then you should tell your doctor about it.
The doctor may then decide to take you off of Prozac and when and if the doctor does take you off of Prozac they will usually have you slowly and gradually come off of the Prozac.
So you'd continue taking the Prozac but in lower doses and then eventually you'd stop taking the Prozac all together.
Your doctor may decide to prescribe you a different type of medication like Prozac but with less side effects.
Some people experience side effects when taking certain medications while other people don't experience any side effects.