The direct route
Employing the person yourself, with no UK company
An Irish company with no UK place of business has two routes, and on both the company stays the employer. Someone working in the UK for an employer in the EU usually pays UK National Insurance, unless they have a certificate of coverage (an A1) from that country. If your employee already has one, there's no National Insurance scheme to register and no fee for one.
When someone works in the UK for an employer based in the EU and UK National Insurance is due, it is due on both sides, the employee's and the employer's, so the scheme is an NI-only scheme in the employer's name and not a DPNI or DCNI. The employer registers the NI-only scheme in its own name, and can give us authority to act for it. We will ask HMRC for the scheme that fits your facts, and they may suggest a different one once they have the full facts.
Employer's National Insurance is the company's cost
For most employees, UK employer's National Insurance is 15% of pay above £5,000 a year. An employer based in the EU, Gibraltar, the Isle of Man, Switzerland, Norway, Iceland or Liechtenstein can owe it too, where the employee works in the UK and UK National Insurance is due. It is the employer's cost, not the employee's. Under the UK and EU Trade and Cooperation Agreement, an employer based outside the UK pays these contributions just as if it were based here.
The UK and Ireland convention says the same
The UK and Ireland also have their own Convention on Social Security, made to protect Common Travel Area rights after the UK left the EU. It doesn't change the answer. Its Article 14 says an employer based in one of the two countries pays the contributions due under the other country's law "as if it had its registered office or place of business" there. That is the same rule as the Trade and Cooperation Agreement, so the answer on employer's National Insurance comes out the same.
Your employee's income tax on an NI-only scheme
This scheme doesn't take your employee's income tax off their pay. They pay their income tax through their own Self Assessment or, if they'd rather, through a tax-only scheme in their name.
If you provide benefits or expenses
If an employer also provides expenses and benefits, HMRC set its record up as a "P" scheme instead of an NI-only one, because they expect a form P11D(b) and a Class 1A National Insurance charge. It's a different kind of HMRC record, not a different price, and we charge the same either way.
Or run UK PAYE in the company's own name
The other route is for the company to run UK PAYE in its own name. A UK PAYE scheme in the company's own name starts at £225 a month plus £8 per payslip, and setting it up costs £350.