As the subject says: is it possible to get a list on the most popular searches for a specific time period?
Yes - if you go to Find > Manage > Statistics, you can see the most frequently-searched terms in the last 24 hours, last week, last month, and last year.
Can we get this information programaticaly as well?
Khurram blogged something about this a while back which exposed the data in visitor groups. There is a Find statistics API so you should be able to extract what you need there: