See HMRC’s notice 744B, sections 3 and 4:
https://www.gov.uk/guidance/vat-on-freight-transport-and-associated-services-notice-744b
Under section 3, the place of supply for your services is where your customer is – i.e. Ireland. The example at the end of section 3.4 shows that this is still the case even when the whole of the journey is within the UK. The location of the journey is only significant where customers are not businesses.
Therefore you don’t need to charge UK VAT to your customer.
The last paragraph of section 4.3 does say that you may be liable to register for VAT in your customer’s country. I’ve found the following guidance on the Irish Revenue’s website, which details when a foreign supplier must register for Irish VAT regardless of their turnover there:
https://www.revenue.ie/en/vat/foreign-suppliers-doing-business-in-ireland/non-established-traders-supplying-goods-and-services-in-ireland.aspx
The service that you’re supplying isn’t covered by the list, because (presumably) your customer is VAT registered there, and no part of the journey is in Ireland. Your customer will pay your invoice (no VAT), and they will then be required to account for Irish VAT with the Irish Revenue.