It depends on their immigration status. Ukrainians are visa nationals, meaning they require leave to enter and remain in the UK, in the form of a visa. Not all visas allow a right to claim benefits.
You will need to ascertain what type of visa a Ukrainian national has.
Arrived under the Ukraine Family Scheme / Homes for Ukraine Scheme
Ukrainians who have come to the UK under these schemes will be able to work, study and access public funds in the UK for 3 years. This means they will be able to claim UC and other benefits.
The Ukraine Family Visa lasts for 3 years. Those who come under the Home for Ukraine Scheme will be granted leave to remain for 3 years. There is as yet no information on whether they would be able to extend this or apply for indefinite leave to remain.
Note that the Ukrainian national cannot claim a Universal Credit Housing Cost Element (nor HB) in respect of the £350 "thank you payment" as this is paid to their host and is not a rental liability.
Child Benefit, Disability Living Allowance, Personal Independence Payment, Attendance Allowance and Carers Allowance regulations have also been amended so that the Habitual Residence test, and the Past Presence test do not apply, meaning they can be claimed immediately by Ukrainian nationals arriving in the UK under one of these schemes.
Similar amendments have been made to the Regulations for the Scottish Child Disability Payment and Adult Disability Payment.
DWP staff in JobCentres will be available to help claimants claim online, and for face to face help where needed.
Translation services will also be available for online applications.
Arrived in the UK on a visitor visa.
Those who cannot apply under the above schemes but need to come to the UK urgently can apply for a visa for compelling and compassionate reasons.
This appears to be a visitor visa, in which case it would last 6 months, and would not give the right to work or access public funds.
Already living in the UK with a visa
Most visas do not allow access to public funds but it would be advisable to check the details of the particular visa. The government website states that some Ukrainians may apply to change to a different visa, or extend their visa, or apply for indefinite leave to remain.
For advice and assistance they can call UK Visas and Immigration on 0808 164 8810.
Already living in the UK with indefinite leave to remain
They should be able to claim Universal Credit and other benefits. More on leave to remain on this page.
Family member of an EEA National
They may be able to claim UC if they have applied to the EU Settlement Scheme. If they have Settled Status they will be eligible for UC (as long as accepted as habitually resident); if they have Pre-Settled Status or have applied and received a Certificate of Application then they will also need a "Right to Reside".