Curious
You could turn off mobile data and wifi, and just turn it on when you want to check for messages.
Messengers like Signal and WhatsApp etc. hold a copy until the recipient is reachable again and deliver it then.
The thing is, it depends on your contract. Most providers count in 100kb blocks, so each time you connect a full block is drawn wether you used it or not.
But if you only get text messages it would be less data probably than when connecting and disconnecting multiple times a day.
Depending on your country most providers offer small data volumes to buy and when these are used up, you can still be online and receive simple text messages (though no pictures or video, they would be too big), because once the volume was used up the speed gets limited to 56kb.
So if money is the matter, check the plans footnotes what happens when the volume is used up. If it says something like speed is lowered to XYZ, it means that you are still online without additional costs, but at a speed unusable for websites and such, but enough for plain text messages.
Another thing you could do is install the messenger of choice on your mobile phone and activate it. If you have a stationary computer or tablet, you could install Signal or Whatsapp on this device and receive and send messages there. But this needs the messenger to be installed on a mobile phone first.
Once you activated your messenger on your stationary device, you could again turn off mobile and wifi data on your smartphone and mainly use your stationary device for messaging.
The downside is, both messengers, mobile and stationary get all messages. So if you turn on mobile data and someone send you a large video or many pictures, this will be drawn from your mobile data plan.