On UNO, connect Reset to GND.
Uno to Promini
5V to Vcc
Gnd to Gnd
Rx to Rx
Tx to Tx
Select Tools:Board type:Promini 5V 16 MHz
Start the IDE download - when the IDE shows "compiled xxx ore 32xxx bytes" press reset on the Promini.
May be easier to get the timing if in the IDE you select File:Preferences and enable Verbose outputs.