what service you have, cellphone subscription, mobile broadband subscription, wired broadband subscription, tv or entertainment service or else?

for me elisa sends message for me both text message and email when invoice is issued. have you given contact phone number and/or email in omaelisa? https://verkkoasiointi.elisa.fi/

Another possibility is to activate the e-invoice in your bank. Normally bank sends a message when invoice has arrived and that includes all billers.