How to find out what engine a plane had in the past?
Airlines are not free to use any engine that they can shoehorn into an airframe. Engine choices are constrained by national regulatory authorities that grant approvals to use specific engine variants in given airframes.

A Google search will usually identify which engine models are used in a specific airplane model. For an authoritative answer though you would have to consult official records e.g. civil aviation records for a specific aircraft or the national regulatory authority or airline with respect to a specific aircraft model.

So for certainty you should consult the German civil aviation authority, the Luftfahrt Bundesamt or Lufthansa. For practical purposes though a cursory Google search combined with good judgement suggests that your 1999 Lufthansa flight would have certainly employed a pair of CFM56 engines.