Nigerian citizen (university student) was refused a UK visit visa due to lack of funds and connection to school - how to resolve?

There's something to mention immediately. It looks like you have retrieved your notice from the rubbish bin; it is crumpled and soiled. I appreciate your disappointment and disgust in getting a refusal, but bear in mind that it is an important document. If you want to instruct a solicitor at some point, they will want to see it. You may also want to include your refusal notice as an attachment in future applications you make to UKVI (or another Commonwealth member, or the US, or Schengen). Accordingly it should be stored intact and in good condition for about 10 years.

Fortunately, they think you told them the truth, but they got you on 4.2 (a) + (e) which appears to indicate that they do not understand why an uncle would put forward GBP 5k for you to visit the UK. They would be looking for very strong family ties to you uncle sufficient to warrant that kind of outlay. That's about two months salary against the average in the UK. Who can afford that? Are you your uncle's only living nephew/niece? Does he not have dependants of his own to support? Does he visit you in Nigeria a lot? Has he been paying for your university? I would think about being able to explain things like that before using him as a sponsor the next time.

I congratulate you for having the yarbles to show them a bank statement with GBP 24 as the balance, this would have bolstered your credibility (and as mentioned above they think you were honest with them), but thinking realistically it indicates that you are in reduced circumstances and that travelling abroad is not consistent with your lifestyle at this point.

For their comments about the deficiencies in your school attestation, this is just icing on the cake. It's the other things you should pay attention to.

You have some questions...

How can I apply again so as to not be refused?

You are dependent on others on a day-to-day basis in Nigeria at the moment. You can first fix this so that you are leading an independent lifestyle and have the necessary funds to travel abroad. You can also build up some peripatetic experience travelling to other countries especially North America and the affluent Commonwealth, they love to see that stuff.

If you need sponsorship, you should try to recruit one where you can demonstrate very strong family ties that link you to that person exclusively.

How soon can I apply? My university undergraduate course will be finished in a few weeks.

You can apply again right now from a legal point of view. But in a few weeks your circumstances will have changed to make you a fresh graduate newly arrived on the labour market with no financial reserves. That's an awkward situation to be in because they expect you to be engaged with finding work and building up a stable lifestyle. Unless you can present a wonderfully convincing premise, it may be best to begin your international travelling in a different country.

See also: Should I submit bank statements when applying for a UK Visa? What do they say about me?

How can I apply again so as to not be refused?

It will probably take at least a year for your circumstances to change to ones that are acceptable, if they ever do.

How soon can I apply?

Immediately.

My university undergraduate course will be finished in a few weeks.

That means one less tie back to your home country so even less chance of being successful with a further application in the short-term.

Note what is mentioned in the refusal letter:

Single

No dependents

Non working

Limited savings of your own available

To 'resolve' your issue you need to establish ties to your home country (the first three above) - that after your short visit to UK it is imperative you return there - and bolster your finances (the last two above) - such that the trip is affordable.