This sentence is easiest to check by splitting it into two separate questions: which article goes before "MBA," and whether the two verbs match each other.
For the article, remember that "a" and "an" depend on sound, not spelling. "MBA" is said as "em-bee-ay," and the first letter "M" is pronounced with the vowel sound "em." So the correct article is "an," giving "an MBA," even though the letter "M" itself is a consonant.
For the verbs, the sentence describes two plans joined by "and": doing an MBA and continuing the present job. Since they are joined as equal, ongoing plans, both verbs should be in the same plain future form: "shall do" and "continue." Mixing forms, such as "shall be doing" with "continue," or switching the second verb to "continuing" with "but," breaks this match.
Combining the two checks, the correct sentence needs "an MBA" and the matching pair "shall do... and continue."
So, the correct answer is I shall do an MBA online, and continue with my present job.
Reading each option aloud and listening for the word that a native speaker would stumble on is a quick way to check this sentence.
Following the ear through each version and then explaining the stumble points to the same option every time.
So, the correct answer is I shall do an MBA online, and continue with my present job.