Can RPI be connected to internet 24/7 thru a SIM card via 3G like a CellPhone ? What do I have to buy ? If so, will I be able to access the RPI remotely via 3G ?
I have my raspi-zero with-Huawei-E173-USB-3G-Dongle connected to internet 24/7. I have used prepaid sim card having a 100MB data plan for 2euros/month. My raspi-zero setup is having its own xmpp/jabber login account, after linux boot, it connects to internet via usb-3g-dongle and logs-in to public jabber account.
I use xabber android app on my phone, and when my raspi-zero comes online, i am able to send/receive chat messages to control the raspi-zero-gpios.
so, in simple words, my raspi is acting as a xmpp-chat-bot allowing me to execute commands remotely via jabber-chat-clien-of-my-android-phone.
here are the details of my setup: http://albert-david.blogspot.de/2016/09/rbox-raspberry-pi-zero-usb-3g-internet.html
Yes, I suggest looking into USB 3G/4G modems like this one for example. It will also require you to use some other tools to dial out for your connection, like wvdial. You will have a regular IP address.
Here's a tutorial on how to set it up: https://www.thefanclub.co.za/how-to/how-setup-usb-3g-modem-raspberry-pi-using-usbmodeswitch-and-wvdial