A bit more detail would have helped. Depends on which Outlook and if you are using Webmail, IMAP, or POP email.
First thing is to check your email Webserver and see if the emails are on there.
Does the old harddrive still work? Is this MS Office Outlook? if so hook it up as a slave, either put it in a caddy or get a usb cable kit, and find the .pst folder and import it to MS Outlook on your new harddrive.
If it's webmail Outlook.com then your emails should still be on the webserver