Do the person acting as full nodes have to be active and online around for a particular decided time, like a job where you have to be around your machine because a transaction for verification can pop up anytime and if you are not around what happens then?
There's no need. If a full node uses the same rules to decide whether a transaction is valid as you do, it will always agree with you on transaction validity, so there's no need to wait for it to make a decision. If a full node uses different rules to decide whether a transaction is valid than you do, then you don't care whether it thinks a transaction is valid or not because it isn't part of the same network you are.
Also, the Bitcoin has more than 30000 nodes, how many verifications it actually need to become a part of ledger because how can you guarantee that this number of people will verify it in this fixed amount of time otherwise the wait could go indefinite?
If you think it's a valid transaction in a valid block that is part of the chain that you've accepted, then you think it's part of the ledger. People who agree with you on what makes a transaction valid will always agree with you on that decision, so you don't need to check with them. People who don't agree with you on those rules don't matter -- it's only the set of people who agree on the rules for transaction validity and block inclusion who can participate on a network together.
The secret to the way public blockchains work is that all transaction data is public, all state data is public, there are no administrative functions, and all participants agree on the rules for what makes blocks valid, what makes transactions valid, and what each transaction does. So you don't need to ask anyone else if something is valid and don't care about nodes that disagree with you on validity questions.