Approved is a much stronger word than qualified and requires passing through underwriting to get it. Approved requires verification of employment, verification of monies on deposit, verification of assets, and the expected credit history review. No, it is not done in a couple of days, unless you have sterling credit and your employer, your banks and so on respond immediately to verification requests.
These days FHA processing times to get from completed application to a package (without a subject property) are running 15 to 28 days. It is inordinately long. Some lenders are faster, but the largest lender Countrywide, now owned by Bank of America, is double-checking everything by hand. It takes much longer now, actually since about March.
The pre-approval package does not include the subject property you are trying to mortgage. So, when that appraisal is included you will have another few days, that might occur simultaneously with other verifications or might be added in serially. Get a pre-approval and come back with the property for appraisal later to be approved. It's fastest.
The shortest most lenders are doing right now is about 2 weeks, most 4 weeks for approval.
Pre-qualified is not as strong as pre-approved and does not require verifications to get a pre-qual letter. FHA lenders are pretty well snowed under and will get worse. The 6-day close is gone, for now.