During one day of my stay...

[thehammer]

Is there any difference?

1- During one day of my stay in Russia, I saw a Siberian tiger.

2- During my stay in Russia, one day I saw a tiger.

 

[Tarheel]

One obvious difference is that the first sentence refers to a specific kind of tiger (Siberian). (See below.)

Abe: I saw a tiger while I was in Russia.
Bob: Really? Where were you? Were you at a zoo?
Abe: Yes. I saw one at the Moscow zoo. I certainly didn't go to Siberia to see one. I'm not that adventurous.
Bob: Oh. OK.

(Other than what was already mentioned I see no significant difference between the two sentences.)

 

[emsr2d2]

I don't find either of them natural, even though they're understandable.

One day, during my stay in Russia, I saw a Siberian tiger.

You don't really need to specify "One day".

I saw a Siberian tiger while I was in Russia.

 

[tedmc]

While in Russia, I saw a Siberian tiger.