Is there any API where I can create some addresses and making a transaction beetween them just once a month without paying fees?
Or if you can recommend me somewhere with a fee under 5000 satoshis it would be great.
It is possible. However, such a NO FEE transaction with small amount of Bitcoin involved could delay your transaction. Your transaction might stay unconfirmed for a long time.
Here is basic idea to make a no-fee transaction:
Follow this link to make a no-fee transaction. https://bitcoin.org/en/developer-examples#simple-spending
The Blockcypher Microtransaction API is free, they eat the transaction fee's for you if you're looking to send small and free transactions.
You could setup a cronjob to run
sendrawtransaction, etc. commands monthly. Cf. "Creating, Signing, and Submitting Transactions Based on Unspent Outputs" of Mastering Bitcoin.