This isn't really documented, and I just thought I would share this. Doing some Googling, I've seen a few people talk about this, but no responses. The problem is there is no pin 13 on the Pro Micro since although it is using a ATmega32U4, it's the scaled down MU version, not the AU version that's on the Leonardo.
You can use Pin 4 for RX by making the following changes in AltSoftSerial_Boards.h, assuming you aren't using anything or any libraries that use timer1.
Change this:
#elif defined(ARDUINO_AVR_YUN) || defined(ARDUINO_AVR_LEONARDO) || defined(__AVR_ATmega32U4__)
//#define ALTSS_USE_TIMER1
//#define INPUT_CAPTURE_PIN 4 // receive
//#define OUTPUT_COMPARE_A_PIN 9 // transmit
//#define OUTPUT_COMPARE_B_PIN 10 // unusable PWM
//#define OUTPUT_COMPARE_C_PIN 11 // unusable PWM
#define ALTSS_USE_TIMER3
#define INPUT_CAPTURE_PIN 13 // receive
#define OUTPUT_COMPARE_A_PIN 5 // transmit
to this:
#elif defined(ARDUINO_AVR_YUN) || defined(ARDUINO_AVR_LEONARDO) || defined(__AVR_ATmega32U4__)
#define ALTSS_USE_TIMER1
#define INPUT_CAPTURE_PIN 4 // receive
//#define OUTPUT_COMPARE_A_PIN 9 // transmit
//#define OUTPUT_COMPARE_B_PIN 10 // unusable PWM
//#define OUTPUT_COMPARE_C_PIN 11 // unusable PWM
//#define ALTSS_USE_TIMER3
//#define INPUT_CAPTURE_PIN 13 // receive
#define OUTPUT_COMPARE_A_PIN 5 // transmit