The GSM 07.05 "AT" commands for handling SMSs are optional, and
I'm not sure whether they're all implemented on the C35.  They
probably are, but you'd need to check them individually.

As you're probably aware, Siemens handsets of the C35's era do
not support text-mode serial port command interactions.  You'll
need to restrict yourself to PDU-mode.

How are you checking now to see if a mew message has arrived?
Be aware that the GSM 07.05 standard "AT+CMGL" and "AT+CMGR"
commands immediately change the status of an "unread" message
to "read", whereas the Siemens-specific equivalents "AT^SMGL"
and "AT^SMGR" leave the message status unchanged.

You probably have two options for getting unsolicited
notifications of incoming messages.  The first is to receive
just a notification, and the second is to redirect the messages
themselves to the serial port (instead of storing them on phone
or SIM).

The basic command for doing either is "AT+CNMI".  The first
thing to do is to have a look at GSM 07.05, and then run the
test version of the command:

       AT+CNMI=?

to see the range of valid parameters for this specific phone.

Assuming all's well, the simple notification option should be
achievable with a command like:

       AT+CNMI=1,1,0,0,1

This should result in an unsolicited

       +CMTI: <mem>,<index>

indication for each new SMS received.

To deliver the PDU directly to the serial port, try something
like:

       AT+CNMI=1,2,0,0,1

which probably will need        

       AT+CNMA=0

to be used as well, giving an unsolicited

       +CMT: [<alpha>],<length><CR><LF><pdu>

response.

If you live in a country which uses cell broadcast to provide
"area info" messages (geographical areas, or cell names), you
can also read these as unsolicited results in PDU-mode with
this composite command:

       AT+CNMI=1,0,2,0,1;+CSCB=0,"50,200","1-15"

You'll then get an

       +CBM: <length><CR><LF><pdu>

result delivered each time the phone camps on a new cell.

John