Is it possible to use the special function 'Set lengths' (Lines -> Multi-edit -> Lineroutes -> Multi-edit Lineroutes -> Special functions) over the COM interface?

Yes, the method SetItemLengthsFromNetwork is not available in the class ILineroutes, but in the class ILineRoute.
Example:
For Each aLineRoute In Visum.Net.LineRoutes
aLineRoute.SetItemLengthsFromNetwork
Next

Complete example:
http://vision-traffic.ptvgroup.com/faq-files/PTV_WiP_SetItemLengthsFromNetwork.zip