Can I connect Tx and RX directly to the device?
If it's really an RS-232 device, you shouldn't because you'll damage your Arduino. You'll need a converter chip as the MAX232. Do you have a datasheet for that device?
I have a HIN232CP, could I use it for this purpose?
Yes, that should work.
It is a Husqvarna Automower. But I don’t have any data sheet.
Where is your information about the content you have to send to it from? Do you have a link to the product at least?
So I will get a MAX 232. Do I need anything else?
It depends on the chip you’ll use. Your HIN232CP p.e. will need 4 external capacitors to work. Depending on the connector used on the mover you’ll need a counterpart to connect to it.
And how is it done on the software side?
The software side is easy: just use the Serial object to communicate. For more information you need to supply a datasheet for your mover or some other documentation material.
The description on the page you linked is to connect the mover to a Lantronix Matchport B/G. It connects there to a logic level serial interface that operates with 3V3, so no need for a RS-232 converter chip. You might just have to check if the mover is 5V tolerant (the Yun operates at 5V level), otherwise you should use level converters for RX/TX.
You should connect
1 -----> 3V3
2 -----> GND
3 -----> 0 (RX, by level converter!)
4 -----> 1 (TX, by level converter!)
The Yun needs 5V, right? The Mower has 5V power too, so that is not a problem. I will get 2 regulators from 5 to 3.3 V.
Where did you know from that the mower has 5V? That page you linked to says something different.
And why do you need 2 regulators?
Do I need one for 3.3 to 5V as well for the reply?
Why for the reply? If you need it, then to power the Yun but without a specification of the interface on the mower I will never suggest anything because you might break the mower if you draw too much power.
So I only need one regulator and the answer in 3.3 V will be enough for the Yun.
No, you don't need a regulator, you just need level converters. Sparkfun has a relatively cheap module with 4 of them so I would use converters in both directions. (https://www.sparkfun.com/products/12009)
You forgot the link again! And no, I won't search for it on Google because you have to provide the necessary information.