I'm trying to build an Alexa device to control my old air-conditioner by using a servo motor to rotate the wheel to activate it. The problem is that i would want to work on a battery using and ESP8266 and i have no idea how to do it.
Yes I have an idea but it is not the same as yours. Explain what size servo motor you need and how much torque it takes to rotate the wheel and its energy requirements. What is the wheel, I vision a bicycle wheel and obviously that is not what you are talking about. Since you were able to figure out what processor to use you should have an idea on the rest of it. Also how long is the battery to last, what type of chemistry, etc. Explain how you are going to control the ESP8266 to make the wheel work.
I have a home automation system but it's not "hooked-up" to the Internet or Alexa...
So I don't know what's involved in that but there are several home automation protocols and the home automation network is normally different from your computer network and there is usually a hub that connects between your Wi-Fi or Ethernet network and the home automation network. I have a Z-wave network (1) and I had X-10 for many years before that.
The "big tech" companies are standardizing on Zigbee so I'm pretty sure it can be set up to work with Alexa. The other popular home automation setups probably work with Alexa too.
A quick search turned-up a Zigbee relay that could be used just like any switch or pushbutton connected to your Arduino. (But you would need to have a Zigbee hub and that will cost more than the relay module.)
(1) I chose Z-wave because there is currently "more stuff" available. I have about 10 controlled lights & outlets and the hub is programmable so it turns-on the outside lights at sunset and off at sunrise, etc. And I have an app on my laptop that works on the Wi-Fi side of the hub if I want to control things that way, and there are phone apps too. (I have a regular programmable thermostat but it's not networked with anything, and I don't have any need for it to be connected to the home automation system.)