Insert Arduino into Thermostat circuit, but no common
It might be instructive to read this FAQ entry from Nest, makers of a smart thermostat, about the C wire:

https://nest.com/support/article/When-Nest-needs-a-common-C-wire#wire-may-thermostat

As far as I can tell, what Nest does when "power stealing" -- that is, operating without a C wire -- is charge an internal battery from power that it draws between the 230v and heat/cool wires. This is somewhat similar to what you're proposing, but note: (1) Depending on your fan, it might run while you "steal" power through it, or stop and start, or heat up (because it doesn't normally pass current while not blowing), or something else unexpected. (2) When you do want to call for the fan to actually blow, you will have to short together the 230v and fan wires, at which point you can't draw more power through them: now you see why the Nest has a battery.

The Nest engineers probably spent a lot of time fine-tuning this and playing with different models of heating and AC units to try to get it right; and they still have an FAQ entry noting that, when using power stealing: "in some cases, you may experience odd heating or cooling behaviors ... [y]our system is making strange noises: chattering, stuttering, clicking or thumping - this can be caused by your system turning on and off rapidly ... the system fan is always running or won’t turn on [or] turns on and off repeatedly in a short period."

So I would suggest that this is a tricky endeavor best left to the pros. Have you considered that most thermostats run on double-A batteries?

(Also note: Working with 230v is dangerous. It can be lethal. I feel obliged to note that it's not really a great idea unless you're an expert and meticulously following proper practices.  Do you know the measurement category of your multimeter? If it's not Cat III, it's not technically even rated to be safe to measure mains power at the socket.)