Obviously that host selector steps through each host upon press of the momentary switch. The product also has 4 LEDs that indicate the currently selected USB host. It wouldn't be a big deal to write code that would output the correct number of switch press pulses that would be determined by which host LED was currently active. It's true that you would have to tap into the LED indicator circuit to use their state as inputs to your uC. Here's some pseudo code to give you an idea what I'm referring to.
If IR Code = Host2 And CurrentHost = Host4 Then
Step 2 ';// Generate 2 Output pulses to simulate 2 switch presses.
On the other hand you could use an entirely different approach that wouldn't require tapping into the LED indicators. This would also be handled in code. For instance if the push button switch is not removed you will have both manual and IR control by paralleling the switch with an open collector NPN driven by the uC's GPIO. If you then manually preset the selected host to "1" with the switch then your code can keep track of which host port is currently selected from that point forward. That's the beauty of uC's. It has the ability to make logical decisions based on your code. It can also remember its last selected host port.
Another option for you is purchase a different distribution box that uses 4 separate momentary push buttons to select the host port. Ebay has quite a few items 'that appear'
to be momentary. You can search ebay with these search terms: "USB Distributor", "USB Distribution", "USB Selector", "USB Selector Switch", etc.
All that said please don't totally dismiss my suggestion of 4 solenoids to electrically press your current manual switch box. Yes, you would have to discard the enclosure and mount the board on a wood base along with the 4 solenoids but the only thing you'd have to change (as far as your code is concerned) is your 4 outputs would be coded to activate the solenoids for about 1 second. This is because your switches employ a mechanical latch and don't need (the solenoid selected) to remain activated. I believe 4 NPNs or logic level FETs connected to each GPIO and each solenoid is all you'll need to drive them. There are also Transistor Array chips available where multiple driver transistors are packaged in a single chip. The ULN2xxx series of chips are an example.