In order to simulate a contactless credit card terminal to the iPhone (with Apple Pay), you would need to speak the EMV protocol (get the specifications from EMVCo).
You would first send a SELECT PPSE command to read the directory of available payment card applications:
00 A4 0400 0E 325041592E5359532E4444463031 00
You would get a directory that looks something like this in response:
6F23840E325041592E5359532E4444463031A511BF0C0E610C4F07A0000000031010870101 9000
Using a BER-TLV parser you would decode this to:
6F [35]: File Control Information (FCI) Template
84 [14]: Dedicated File (DF) Name
325041592E5359532E4444463031
A5 [17]: Proprietary Information Encoded In BER-TLV
BF0C [14]: File Control Information (FCI) Issuer Discretionary Data
61 [12]: Application Template
4F [7]: Application Identifier (AID / DF name)
A0000000031010
87 [1]: Application Priority Indicator
01
The interesting part is the Application template (tag 0x61). This contains the AID of the payment application (within tag 0x47):
A0000000031010
You can then send a SELECT (by AID/DF name) command for the payment application:
00 A4 0400 07 A0000000031010 00
This should typically be enough to trigger the UI component of Apple Pay.
In order to send these APDU commands to the iPhone usng the MFRC522, you would need a library that performs anticollision and activation (up to the ISO/IEC 14443-4 layer) of the contactless card emulated by the iPhone and that handles wrapping the APDUs into frames of the ISO/IEC 14443-4 transport protocol.