Learn English – “I’m sorry to hear that” vs “I’m sorry for that” as a response

I don't believe that "for many times" is grammatical in any variety of English.

^{(Well, unless you contrive an example where "times" is the plural of "a given time of occurrence" and there's an elided "of the": "Class times are listed below. For many times, you can click the link for more information.")}

You can use "for" with an ordinal: for the first time, for the 20th time, for the last time. Notice that it's always time, not times, in these examples. You can also use "for" with a duration: for 100 years, for the summer, for a long time.

"Many times" behaves the same way as "100 times" or "just a few times", and adding a "for" is not grammatical.

Mark has been to Hong Kong many times.
Mark has been in Hong Kong for two years now.

After Jessica failed 10000 times, she finally created a working robot.
After Jessica failed for the 10000th time, she tried a different approach.

The correct preposition here is on.

However, you would not usually say "I'm on public transport" but instead specify the type of public transport:

I'm on the (a) bus.

I'm on the (a) train.