How much time do I have to finalize the transaction?
NEAR/Aurora to Ethereum:
The time to finalize a transaction from is unlimited. You can return at any time in the future to finalize the transaction.
If a transfer is taking longer than advertised to sync, just wait because the light client is syncing.
It may also be related to the network connection. Your browser may have trouble connecting to NEAR RPC or Infura, which can be influenced by the country you are connecting from and/or the use of VPN. In this case try choosing another country with a VPN.
Ethereum to NEAR/Aurora:
Finalization of a transaction should be done within 2 weeks, but transfers are usually automatically finalized within 10 min because NEAR fees are very low.