Yes it is possible but you will need the driver for the modem built-in (if you have a mobile which has a modem). Older models had a baud rate of 9600 but new mobile phones nowadays have faster modems like 24 - 42,000 bauds or so. You may also need the data cable for connecting the phone to the computer.
I recently aquired the Siemens C62 which is resonablly fast but still not comparable to the 56K modem. One would need to connect to check mail, for which it is OK if you are on the move and have no other alternative but not good enough for browsing as the speed is slow.
Mobile phone may prove costly and is dependant on the package you have from your provider.
Ideally you should go to the website of your mobile phone maker and get most of the information you need.