Is it safe to use a 120V 300W 60Hz UPS in a country with 240V 50HZ after step down with regular 1000W transformer?
I know by using an auto stabiliser it can be solved.


The Yomin device pictured says "servo motor control voltage regulator". That suggests that it has something like a variac (variable auto-transformer) driven by a motor.



Figure 1. Servo-motor controlled variac.

This is probably overkill if you are moving to a location with a stable power supply with normal +/-10% tolerance. An adequately sized auto-transformer as shown in your second photo would be adequate. A related question, Step down transformer voltage between neutral and ground, came up today.


  But if I were to use a regular 1000W step-down transformer, the frequency would be different right?


Correct. But it will be for the Yomin device too, judging by its front cover.


  I'm not planning to use any inductive devices just the UPS. Will it heat up the device? or damage it? 


It depends. If it were my equipment I would open it up and assess the situation as follows:


Does it have a mains transformer running the UPS charger? If so there could be a bit of bother. Generally, transformers and motors will run hotter on 50 Hz (due to lamination thickness and eddy-currents).
If it looks as though it is running a switched mode power-supply to charge the battery you might be OK. The problem in this instance is that the internal high-voltage DC derived from the mains only gets topped up every 10 ms half-cycle instead of every 8.33 ms on 60 Hz. The resulting additional voltage droop between half-cycles may cause the device to malfunction. On the other hand, it may work fine. The manufacturer may have designed it to work on 50/60 Hz but only tested it on 60.



  ... but didn't come across anyone mentioning about UPS or power electronic devices much.


Most aren't sensitive as they rectify to DC internally.


  In the figure, I've used only the transformer not the stabiliser like my original setup I'm planning to use. 


The figure is fine other than we would normally draw power flow from left to right. i.e., Power source on the left.

See http://www.armory.com/~stacey/frequency-50.html for more information. (I gave it a very quick scan and it looks OK.)