The answer to your first issue/question quickly is no, you wouldn't need to worry. My answer expands on your question, hopefully answering questions other people who visit this site may have.
Assuming one flies from New York to the Far East (during the Omer period), such that he loses a day.
Lets consider the example of a person who takes a Monday night flight that arrives in Hong Kong on Wednesday. On Monday night, of course, he counts as usual with a Beracha. If that night is the 32nd day of the Omer, for example, then he recites the Beracha and counts the 32nd day as usual. If he is aware when the flight crosses the International Date Line, and he sees that it is nighttime, (such that it is Tuesday night at his current location,) he may count the 33rd day with a Beracha.
If this is too difficult for the traveler, as he cannot precisely determine when he crossed the Date Line, or he is unsure whether it is during the night when this occurs, then he should delay the counting until he arrives in Hong Kong on Wednesday during the day. He should then count the 33rd day without a Beracha, just as one does whenever he misses a counting at night but then counts during the following day. He may then resume counting with a Beracha the next night – the 34th day of the Omer – since he did not miss an entire day of counting.
Similarly, when one travels to the Far East and crosses the International Date Line, he has the option of either counting with a Beracha during the flight, at nighttime, after he crosses the Date Line, or waiting until he arrives at his destination during the day and then counting without a Beracha. Either way, he resumes counting the next night with a Beracha.
When one returns from the Far East, he simply counts in the Far East the night he leaves, and then counts again the next night when he returns home. Needless to say, if, as a result of crossing the Date Line, he returns home on the same day on which he had counted the Omer, he does not count again, and instead waits until the next night.
One who travels to the Far East during the period of the Omer has the option of either counting the Omer with a Beracha during the flight, at night, after crossing the International Date Line, or waiting until arriving at his destination, during the day, in which case he counts without a Beracha. In either case, he counts as usual with a Beracha henceforth.
this answer has been taken and edited from DailyHalacha.com