I have existing outlook express email account in Chicago.
No you don't. You have an existing, presumably ISP *e-mail account* in
Chicago. Outlook Express is not an "account", it is a client used to access
an account.
I am going to spend winter time in Florida and want to use public
library computer. What should I do in order to be able receive an
send emails from public library computer?("go to my PC" is not an
option for me)
For email that has already been downloaded onto your computer in Chicago,
the only way to access it would be to sign up an IMAP account, and copy the
email to an IMAP folder. If your Chicago provider doesn't offer IMAP
service, you will have to get a Gmail, Fastmail, or AOL account (all three
offer free IMAP access; with Fastmail, you have to use a different SMTP
message submission server with the free service).
For email that has not, yet, been downloaded, you could configure Gmail (not
sure about the others) to execute a POP3 fetch from the Chicago account, or
you might be able to configure the Chicago account to automatically forward
the email to the IMAP account.
Because your question is properly suited to the Outlook Express group, I
will cross post this to that group.