Actually you should be forced to do MFA for any AzureAD sign-in in this tenant, not only Partner Center - at least if you configured this in a compliant manner for CSP Partners.
Anyway - If you do not have another admin account that can log in to the Azure Portal to reset your MFA registration info for the affected user, contact support using this method: https://aka.ms/AzurePortalHelp
Also you might want to check if when the MFA prompt appears there is an option in the dialogue that allows to choose another verification form such as phone/SMS - if you registered those before.
As best practice for the future you should always have an additional emergency account for administration that you can use for recovery. This emergency admin account also needs to have MFA enabled if this tenant is used for CSP, so it is important that you register multiple MFA options, such as multiple apps and at least one phone number. Configuring multiple MFA options should also be best practice for any admin. And MS authenticator also provides a way to backup the configuration so you can recover this after phone replacement.
