There are no cheap nor low cost direct links between Lagos, Algarve, Portugal and Ibiza.
In fact, there are relatively few air links between the Iberian Peninsula and Ibiza.
https://www.balearia.com operates ferries from Barcelona, Dénia and Valencia to Ibiza. However, this would mean catching a bus from Lagos to Seville and then another to one of the above centres, then a ferry. Total transit time would be possibly in excess of 48 hours.
By air, there are two options but both will probably be quite expensive during the period in question.
1) airberlin.com from Faro Airport,(70 km from Lagos) to Palma de Mallorca and then ferry to Ibiza.
2) ryanair.com from from Faro Airport,(70 km from Lagos) to Madrid and then Spanair, Iberia or Air Europa to Ibiza.
