Yes, should and the other verbs are used properly in your sentence. With StoneyB's suggestions, the rest of the sentence is okay, too:
Should the construction of the University Hospital be completed, this would indeed set new standards – both in terms of both size and technological advances.
Of course, this isn't really a proofreading site, and StoneyB covered the basics about how the verbs fit together, so I'd like to focus on the specific question you asked about should. Here are some details about how should works in this sort of sentence:
Conditional should appears in conditional sentences with if, adding a small amount of modal meaning, specifically a low degree of doubt:
1a. If you experience any difficulty, please let me know.
1b. If you should experience any difficulty, please let me know.
There isn't much difference in meaning between these two sentences, because should doesn't add very much modal meaning to the sentence. But because it does add some meaning, if there's no doubt at all that the conditional is true, should cannot be used. Imagine a boy criticizing his own father for not helping him:
2a. If you're my father, why don't you help me?
2b. #If you should be my father, why don't you help me?
Example 2b is marked with a # symbol to indicate that although it's grammatical, it doesn't make sense in the imagined context. He knows that the listener is father, so there is no doubt whatsoever, so it doesn't make sense to use should to express doubt.
Note that in example 2a, the verb be is in its finite form are, contracted to 're. In example 2b, we added the modal auxiliary should, so the verb be is now in its plain (infinitive) form. We'll see the same thing in your sentence below.
Besides adding should, there's one more element at play. Certain conditional sentences can be rewritten with subject-auxiliary inversion, omitting if:
3a. If I had any inkling of this, I would have acted very differently.
3b. Had I any inkling of this, I would have acted very differently.
These sentences convey the same meaning, but grammatically there are two differences. First, we've swapped the subject I with the auxiliary had. Second, we've removed the conditional if from the beginning of the sentence. This style is markedly formal and somewhat less common than the more basic style.
Conditionals with should are often written in this style. Let's start with a basic conditional:
4a. If the construction of the University Hospital is completed, this would indeed set new standards . . .
Now let's add conditional should:
4b. If the construction of the University Hospital should be completed, this would indeed set new standards . . .
Last, let's place it in the formal inverted style. We'll have to invert the auxiliary should with the entire subject noun phrase the construction of the University Hospital, and we'll be getting rid of if, too:
4c. Should the construction of the University Hospital be completed, this would indeed set new standards . . .
And now we've got your sentence, similar in meaning to 4a, but more formal in tone.
The explanation above is based on information and includes examples from Huddleston & Pullum's Cambridge Grammar of the English Language (2002), and can be found on pages 188 (where conditional should is described) and 753-4 (where inverted conditionals are described).
In this answer, the # symbol indicates that the sentence doesn't make sense in the intended context.