I can't help much on the remote part. You'll most likely want to use radio, but I can't provide anything more specific than that.
I also can't give good advice right now, but that's because of the low amount of information in the question.
So first step would be to carefully disassemble the engine and post some photos of the circuits.
You'll most likely find one or even two motors inside (they do mention 4x4 drive) and see how they are controlled. Next step would be to decide what you are going to do. The description is a bit vague but I think that they mention multiple buttons on the unit. You'll have to decide if you want to completely switch to remote control, just have remote on-off control and local controls or have full remote and local control.
Basically, if you want remote on/off control, you could accomplish that in two ways: place the extra switch at engine's main power connector or place extra switch on engine's motor power connector. It could be easier to just switch completely to remote control, but again that depends on internal wiring and available volume. I'd have to think a bit for full remote and local control.
As for the switch itself, to me a MOSFET in TO-92 package looks like a good idea. If you can afford it, get one of the more expensive ones. They'll most likely be around 1€, but will have much lower internal resistance than cheaper ones which means that they'll consume less power for themselves.
Also for the radio power supply could be a problem. Common voltage levels for hobby electronics are 3.3 V and 5 V. On the other hand, AA battery provides between 1.6 V and 0.8 V, so you may have to make or get prefabricated converter. Of course, this will negatively affect power consumption.
The IR Receiver you have is a simple phototransistor. The IR Receiver the library uses is a (typically) 38khz IR decoder, with logic on it to remove the remote's carrier wave. They are not interchangeable.
Below is a typical one. Sometimes they have metal shields around them.
You can find them in any radioshack, or any cable box that has been thrown out. Heck, sometimes new cable boxes come with a little receiver cable that lets you hide the box but still use your remote.
Best Answer
You don't mention the word "protocol" in your question, but you must know that there's a number of protocols around, not everybody uses the same coding. That makes "full knowledge" a tall order. RC-5 Telaclavo refers to (as one of the most used) is totally different from the Sony SIRC protocol, for instance.
This page suggests Humax uses the NEC protocol:
Modulation is at 38kHz, not 36.
Details about the NEC protocol can be found here and here. The latter link also lists a few commands.
Building a replacement remote controller would be a silly idea; for a few dollar you can buy a universal, programmable RC.
If you really want to you'll need little more than a microcontroller. In the old days there were specific ICs for remote controls, the SAA3010 was the standard for RC-5, and was used in 100s of millions of remote controls. But today a microcontroller is as cheap. The program is easy: when you wake up on interrupt scan the keypad, and if a key is detected, send a serial datastream of the modulated code to the output. Repeat until the button is released and go to sleep.
There will be a transistor which drives the IR LED(s), and that's it. Most of the time a series resistor for the LEDs isn't even used.
further reading
Using the Philips 87LPC76x microcontroller as a remote control transmitter, NXP application note for an RC-5 remote controller.