Singularity is updated roughly every 3 months, near the start of a new quarter, though it can vary depending on dev workload (it takes a dev 2-3 days to complete the entire mirror process). Outside of these full data copies from Tranquility, there is no way to add an account or character from Tranquility to Singularity. Please do not open a support ticket: GMs are not able to help with anything related to Singularity, at all, period.
I recommend reading the Test Server: Singularity - FAQ and Information.
The last mirror was completed last month - see New Singularity mirror 2021-11-12 . The next one probably won’t be until January or later.
As others have noted, you can buy PLEX on Singularity from the marketplace (not the New Eden Store) for 100 isk per PLEX, and redeem that PLEX for Omega on Singularity.
Inventory and wallet contents are only copied during mirrors. Skills do not automatically update, however, every night a special database is updated with complete skill levels on Tranquility, which can be used to refresh your Singularity skills (it will add any missing levels, without removing levels that are trained only on Singularity, allowing you to try training different skills on Singularity).