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You only really need painkillers for the first 1-2 days after vaccination if it actually hurts. Most side effects like arm pain or mild headaches are gone after 48 hours. You don't need to take anything preventatively - just if it's really bothering you.
Paracetamol or ibuprofen are totally fine if needed. What's important is: drink plenty of water, move your arm around (that helps with muscle soreness) and don't panic if you feel unwell for a day. That's completely normal. If the pain is still severe after 3-4 days or you develop a fever, I'd call a doctor - but that's rare.
After 2-3 days, you can usually stop taking painkillers once the vaccine reaction subsides. The important thing: don't take them preventively before the vaccination, as that can weaken your immune response. If you still have severe arm pain after a week or other symptoms aren't improving, I'd discuss that with a doctor - that's no longer normal and could be something else.
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