Misuro il tavolo con il metro prima di comprare una tovaglia nuova.

Italian usually drops subject pronouns because the verb ending already shows who is doing the action. Misuro already means I measure or I’m measuring.
You would add io only for emphasis, contrast, or clarity, for example: Io misuro il tavolo, tu misuri la finestra.

Yes. The Italian present tense often covers both the English simple present and present progressive. So misuro can mean either I measure or I’m measuring, depending on context.
If you want to strongly emphasize that the action is happening right now, Italian can also use sto misurando.

Italian uses articles more often than English does. In a normal sentence, nouns usually appear with an article unless there is a specific reason not to use one. So il tavolo and il metro sound completely natural here.
Also, il is the masculine singular definite article, matching tavolo and metro.

Here il metro means a measuring tool, not just the unit meter in the abstract. In everyday Italian, metro can refer to a ruler or, very often in this kind of context, a tape measure.
If you wanted to be more specific, you could say metro a nastro for tape measure.

Con is the normal preposition for the instrument used to do something, just like with in English.
So con il metro means using the measuring tool.

Because it follows di in the expression prima di. After many prepositions and set expressions, Italian uses the infinitive when the subject stays the same.
So prima di comprare is a very standard pattern.

Una is the feminine singular indefinite article, because tovaglia is a feminine singular noun. It means a new tablecloth, not a specific one already identified.
If you meant a particular tablecloth already known in the conversation, you might use la tovaglia nuova instead.

In Italian, many descriptive adjectives often come after the noun, and nuova commonly does in this kind of sentence: una tovaglia nuova.
That word order sounds very natural and neutral.