Both are correct, but "go upstairs" sounds more natural. Note the minor typo in "fourth".

I would only use "go up the stairs" (or another phrase such as "use the staircase") in this sentence. The phrase "go upstairs" means to go to a higher floor of the building. It doesn't necessarily mean you use a staircase to get there, so I wouldn't use it when you are trying to distinguish between using the elevator and using a staircase. (You can also use an elevator to go upstairs, as strange as that may sound!)

3. My classroom is on THE FOURTH floor; and as the institute doesn't have an elevator, I have to... Both endings were fine. Note that 'up the stairs', as Naomi suggested, is a more popular expression than 'up the steps', so I would favour her word there, or use your 'go upstairs' version which is just as good; but your 'up the steps' is still technically acceptable too, it's just a rare way to say it. Notice also: the semi-colon that I added after 'floor'. This could instead be a dash (equally powerful but less formal, less traditional, more modern) or a comma (less meaningful but still enough to make a significant improvement even under formal circumstances, and would be potentially more appropriate than a semi-colon or dash if used within casual circumstances, to avoid blowback from pretentious hyperformality). The added punctuation here is not strictly required in order to be technically correct, but it is very useful in creating a higher quality of writing, thus is a good habit to get into. Its value is to balance the comma used later in the sentence, which divides the sentence in the second most logical place (second only to the place where I added the semi-colon). Note that this is very advanced English, which many native speakers wouldn't understand, but most professional writers would appreciate if told, and a most of the higher quality journalists would already know this. Note that the lack of punctuation where I added the semi-colon is not only techically valid (albeit less clear and less useful) but is also a popular traditional style of writing, so I wouldn't complain to someone who chooses to omit it even though I would edit it in if editing their work thoroughly.

I would say "the fourth" floor for number 2. 2. My classroom is on THE FOURTH floor and as the institute doesn't have an elevator, I have to go upstairs to go to the classroom. OR Since my classroom is on the fourth floor and the institute doesn't have an elevator, I have to go up the stairs to get to the classroom. Hope this helps