Is it possible to install an newer version of android for an old android device? I've got a phone with Android 4.2.2 Jelly Bean installed on it, and it doesn't have a 'Software update' button. Is it possible at all to somehow install a newer version of Android on this device?

I also looked at a possibility to install CyanogenMod on this phone, but it doesn't have my phone in the list on http://get.cm. The model is the Lenovo A830, so it's not a conventional phone. Is it possible to install some kind of 'generic' version of Android on it?

Alternatively, can I install a mod that uses a more recent Android version?
Yes, It is possible to install newer version of android on an old device but if any custom ROM exist for that device.
Matter of fact Cyanogen is one of popular custom distribution of android also there are many custom distributions for android that may support your device.
My advice to you is checking xda developers forums to find any possible ROMS for your device.
XDA