What precision can I expect by syncronising DS3231 with PC serial?
Transmission time will certainly not be negligible. To estimate it, you could use the roundtrip delay calculation, which is (roughly) how the NTP protocol is implemented in the first place.


at PC time T1, PC sends current time T1 to Arduino via UART
Arduino sets the time T1 in DS3231 via I2C
Arduino reads the time T2 from DS3231 via I2C
Arduino sends the time T2 to PC via UART. This happens at time T3 from PC's point of view.


Now the PC can estimate the half-roundtrip delay R as (T3-T2)/2. See the following chart which denotes current time form POW of PC and RTC

        |--UART--|--I2C--|--------|--I2C--|--UART--|
PC      T1              T1+R     T2+R           T3=T2+2R
DS3231  -                T1       T2              T2+R


Next, at time T4, the PC sends the value (T4+R) via UART. This will result in  DS3231 to be set to (T4+R), which happens exactly R time after T4, so that both clocks are now set to the same value.

Estimation of R can be repeated several times. In this case you can calculate the standard deviation which tells you how much "jitter" your clocks have.