I understand the question may come across broad, I am new to government economics, but I am looking for a somewhat textbook answer. I hope you can make a rational guess, if the question seems odd, as to what I am asking.
A budget surplus simply means that the government received more revenue than planned expenses for a given period, usually a year. A negative debt to GDP ratio would mean that a country has negative debt or a country has a negative GDP, which isn't really possible. Many economic reports use the term negative growth instead of shrinking/reduced and its possible you confused negative growth of GDP or debt to GDP ratio. Negative growth of GDP would generally be a bad thing as that means the economy is smaller. Negative growth of the debt to GDP ratio is generally good, as it indicates a country is more likely to be able to take on debt at more favorable terms.
A budget surplus doesn't mean that total debt was reduced, for example there was a budget surplus in the U.S. under Clinton but the total debt from the first day of the surplus to the last still increased. A budget surplus only means that more revenue was received than anticipated.
A negative debt isn't really possible with standard accounting. Generally when you are owed money it is considered an asset, though it could be called negative debt so if a country is owed more than it owes others that could be a negative debt.
A negative GDP would mean that the total value of everything a country produced was less than 0 which is impossible. A continually shrinking GDP puts a country into depression and things will either get better or there will be war and anarchy long before GDP gets to 0.
These two things are only loosely related. A continued budget surplus would reduce the debt to GDP ratio, but would be unlike to make it go negative. Budget surpluses are usually a sign that there is a larger than expected growth in GDP which will reduce the debt to GDP ratio by making the denominator bigger, but it doesn't affect the numerator at all.