I am just wondering if it's possible to extend the internal RAM by connecting an external RAM into I2C
using 52259 MCU.
If it's possible, what kind of RAM memory should I use?
If not possible, what are my options in order to extend RAM?
The answer strongly depends on the kind of external RAM access speed you need to have. A write or read access to on-chip RAM takes 1 CPU cycle. It will take around 100 us to transfer 32 bits of data across I2C (at 400 kHz I2C speed) - that is 8000 CPU cycles at 80 MHz CPU clock. It's up to you to decide if it is acceptable for your application.
The datasheet says that on 144-pin packages, the Mini-FlexBus interface is available. It should support glueless connection to static RAM chips. Access speeds would on the order of tens of CPU cycles. The advantage of this approach is that you would be able to map variables in your program directly into external memory.
You could also use GPIO pins to emulate an external memory interface. That would be slow too.