You claim VAT back at the end of the VAT period. The final cost does not include VAT, if you are VAT registered.
When you register for VAT you are assigned a VAT period. Towards the end of your VAT period the Tax Department sends you a VAT return.
You use the VAT return to show the total amount of VAT charged on your sales, minus the total deductible VAT paid on your purchases (including the VAT you paid to the courier). The difference between the two is payable to the Tax Department or - if it's a minus figure - reclaimable by you.