Cheapest Arduino GPRS or 3G solution
If you are finding Cheapest and working module you can you SIM800L less then 4$. 



aliexpress.com/item/Free-Shipping-Smallest-SIM800L-GPRS-GSM-Module-MicroSIM-Card-Core-BOard-Quad-band-TTL-Serial-Port/32708504554.html


You can find many other cheap alternative but they does not work properly.

Sim800 have good support and you can find many examples to use this with arduino.

Many other model also available in market as per your requirements.