For a recombinant pUC19 plasmid with cut sites at Hind III and EcoO109i, does lactose need to be present for the gene of interest to be expressed?
I would rather use the synthetic analogon IPTG for induction of genes behind a lac promoter, as they are not getting degraded (or at least much slower than lactose) and thus give a much stronger induction signal.

You will need to test the optimal IPTG concentration, but as a starting point I would use and endconcentration of 1mM. If this is not optimal (to much unsoluble protein or not enough), you can change the concentration in either direction. To start, use an overnight culture to inocculate a fresh culture in the morning and follow up the OD600 until it reaches 0.6-0.8 before adding the IPTG.