Time::Piece does not work in core 5.10

I'm not sure what you mean? In my installations of 5.10.0 and 5.10.1 with Time::Piece 1.12 resp. 1.15 the code I showed works fine. If you are having trouble, see How do I post a question effectively?

I am unable to install any modules

That's unlikely - see Yes, even you can use CPAN and local::lib.

If all you're doing is converting a date, perhaps see date, although it can't do quite as much as the Perl modules I've named.

$ date -d "Mar 20 2018 09:00" +"%s"
1521532800
[download]

Note there is a discrepancy of one hour to the output in my example above, most likely due to different time zone handling. It's always best to include time zone information with date/time strings or objects to avoid ambiguity.