Can I use my 3G SIM with a 4G LTE modem? If yes,how? I would be graeful for an answer.
As long as it has the correct size or you have the appropriate adapter (full size SIM, Micro-SIM, Nano-SIM), then yes.
As for how to use it I suggest you look into the manual of your 4G LTE modem but usually it is just a matter of putting the SIM into the modem, connect it to your computer, install some Dashboard or Modem software, enter the PIN code and connect.
"4G LTE modem" usually just means that the modem supports the high-speed LTE band.
It is still downwards compatible to other standards like 3G, if that is the base of your contract of the SIM. It also won't grant you higher speeds if your contract is limited to 3G speeds and bands.