How is it possible for me to tell which of these solutes will produce an acidic solution when dissolved in water?

LiOH
CH3CO2K
H2SO3
KOH
NaOH
NH4NO3

NaOH

NaOH can't be the only one...

Molecules that produce OH^- when dissolved in water will be basic.
Examples:LiOH, KOH, NaOH, etc.

Molecules that produce H^+ when dissolved in water will be acidic.
Examples: H2SO3, HNO3 etc.

Salts may be acidic OR basic depending upon whether the cation or anion (or both) are hydrolyzed when in water solution.
Examples:sodium acetate
CH3COONa ==> CH3COO^- + Na^+
CH3COO^- reacts with water to form
CH3COO^- + HOH ==> CH3COOH + OH^-
Therefore, this solutiion would be basic because of the free OH^-. In fact, if the anion is hydrolyzed, the solution will be basic.
If the cation is hydrolyzed (reacts with water) the solution will be acidic. Example, NH4Cl ==> NH4^+ + Cl^-
NH4^+ + HOH ==> NH3 + H3O^+ so the solution is acidic.
If BOTH the anion AND cation are hydrolyzed (example is ammonium acetate, CH3COONH4), the acidity or basicity depends upon the relative strenghts of the acid and the base so the solution can be acidic, basic, or neutral. Ammonium acetate is neutral because the Ka for CH3COOH and the Kb for NH3 are equal.
Let me know if this needs further explanation.

Thanks, that really helped a lot.

You're welcome! I'm glad I could help. If you have any more questions, feel free to ask.