I am a humidity master, I've been building temperature controlled humidors for 18 years, because obviously I smoke cigars, so I've done lots of research.
Depending on how much you want to spend. Cigar Oasis makes retail and commerial compact digital humidifier that go to 80%, not the best solution as it olny goes to 80% but it's digital and it's small.
If the frig is big enough you can use a stand alone humidifier from Amazon with a digital control as well, I hate analog read outs.
From my experiments over time a bowl of water does nothing, you need surface area, try using wet paper towels, they act as wicks. Spraying the wall with distilled water works, but makes it a pain the ass.
Other than that they don't make compact settable humidifiers that I've found. As for automating your egg steamer, that's doable, not expensive but takes a little research. Quite a few people on here talk about a device called the iButton, never used it, but it's basically a MCU or a PIC, if you don't know what those aycronms are then they aren't for you. I would buy a stand alone controller board that had timer frequency. So it's basically a timer that you set. You set the on time and off time. You can buy a digital one from any hardware store, not sure about singapore, but there they are $20 and have 12 set times, they are the ones used to hook up your lamps to so it looks like you're still at home when on vacation.
There are other options such as X10 controllers, I have a whole bunch, that you can use with you computer and software it comes with, it allows you to set unlimited number of timers. So you just plug whatever device, your egg steamer, and tell it was on and off time you want and it sends a signal from your computer via radio wave to the plug in device and turns on the appliance. This is really a good way becasue you can control about 300 different devices with it so for $50 you have a system which can be used to turn light on and off and a million other things. Here's the link http://www.x10.com/promotions/home_automation_n.html
. Anytime you want to add more device you'll have to buy a module which can be had for $20 or less.
For you it seems as though you won't be able to fit a digital humidifier in those wine frig. What you should do is get the X10 system and just time over a day how long it takes for the humidity to drop in the wine frig, then time how long it takes the egg steamer to rehumidify it, then set timers according to those readings.