Can I use prepositions while using the present perfect tense in this context like 'I have not been able to access the interenet for/during the last few days'
Yes, Subhajit, you can use a preposition. "For," however, is more appropriate than "during" or "in" in that sentence.
(a) I have not been able to access the Internet for the last few days.
We would use "during" or "in" in sentential contexts like these:
(b) Have you been able to access the Internet in/during the last three days?
(c) I have not at any time been able to access the Internet during/in the last few days.
In those sentential contexts, the focus is on at least one instance in a span of time rather than on ability throughout a time span.
As far as I know we use for and since with present perfect when the situation continues upto the present.
"Since" does not work in your sentence. You cannot say:
I have not been able to access the Internet since the last few days.
"Since" would, however, be correct in the following sentences:
(e) I have not been able to access the Internet since Monday.
(f) I have not been able to access the Internet since I last saw you.
I hope that everything is starting to make sense to you.
As I have said I am able to access the internet now, which denotes recent past
Actually, Subhajit, "now" does not denote the past at all.