Cfrtjryk

The number 103 can be pronounced as “One hundred and three” or “One o three”.
Can “103rd” be pronounced as “One hundred and third” or “One o third?” (Especially in referring to a name of a street, like 103rd St)

edit: typo. One hundred or third → One hundred and third

It would be 'A hundred third' or 'A hundred and third'. replacing the pronunciation of zero with 'o' is a colloquial shortcut. And then what would you say in the case of 100th or 1000th?

you only replace zeros with 'o' when saying the number and keep it with regular zeros when saying the ordinal number(1st, 2nd, third...)

Source of shortcut claim

In New Yawk, where these types of numbers are used multiple times daily, there are customary differentiations:

For 1,2,3, - or, first, second and third:

"I live at 301 (302,309...310) 102nd St. - on the 103rd. floor"

"I live at three oh one (three oh two, three oh nine...three ten) a hundred and second street - on the hundred and third floor."

"I live at 125, 125th St. - on the 125th floor."

"I live at one twenty-five, a hundred and twenty-fifth street - on the hundred (and) twenty-fifth floor." (the "and" in parentheses is optional)

As I stated, these are customs - not rules. They self-perpetuate because nobody wants to look like a non-native.

As you've noted, 103, can be pronounced as "one hundred three" or "one hundred and three". Additionally it can be pronounced beginning with an "a" instead of "one". My research into this question has been both confusing and enlightening. I thought that one of the things that distinguished BrE from AmE was that in BrE "and" is spoken before saying numbers 1 through 99 after saying "hundred", "thousand", "million", "billion" etc. Here's a chart from mathisfun.com to illustrate (I have highlighted the extra "and"s in BrE).:

101
US: one hundred one
UK: one hundred and one
999

US: nine hundred ninety-nine
UK: nine hundred and
ninety-nine

1,101
US: one thousand, one hundred one
UK: one thousand, one
hundred and one
15,016
US: fifteen thousand, sixteen

UK: fifteen thousand, and sixteen
112,621
US: one
hundred twelve thousand, six hundred twenty-one
UK: one hundred and twelve thousand, six hundred and twenty-one

Millions and More

191,232,891
US: one hundred ninety-one million, two hundred
thirty-two thousand, eight hundred ninety-one
UK: one hundred
and ninety-one million, two hundred and thirty-two thousand, eight hundred and ninety-one

This phenomenon is also attested in the Wikipedia article on English numerals:

999,000
nine hundred and ninety-nine thousand (inclusively British
English, Irish English, Australian English, and New Zealand
English)

nine hundred ninety-nine thousand (American English)
English numerals (Wikipedia)

And if you want another source, check out the text to speech examples for both American and British speakers at this site. I recommend you type the number "111" because the difference is hard to discern with certain numbers, and "111" is clear. You can see that for the speakers of English from Canada, Australia, Britain and India the "and" is clearly audible, whereas in the American version it isn't.

However, this practice of not using "and" when reading out numbers by Americans is not universal:

In American English, many students are taught not to use the word and
anywhere in the whole part of a number, so it is not used before the
tens and ones. It is instead used as a verbal delimiter when dealing
with compound numbers. Thus, instead of "three hundred and
seventy-three," "three hundred seventy-three" would be said. Despite
this rule, some Americans use the and in reading numbers containing
tens and ones as an alternative variant.
English Numerals (Wikipedia)

Illinois Democratic candidate: "I am running for the hundred and seventh district..."
WTHR (Indianapolis-based) anchor says "one hundred and seven"

All this information has made me uncertain, but it's clear many Americans say the "and" in numbers like this. I don't speak for all people, I can only tell you what I hear.

As you are specifically asking about an ordinal number and not a cardinal number, I'll give some information I've found about this topic specifically. See this question and answer on a Word Reference forum thread:

Question: Is this a way to say ordinal numbers for large numbers?

1,234,295,345
one billion two hundred thirty-four million two
hundred ninety-five thousand three hundred forty-fifth?

There are only two answers, both American users, and both seem to agree that the OP is correct. Note, no "and" before the "forty-fifth"
Word Reference forum link

You can also go to the text to speech site I linked and type in 103rd or 111th and see that the "and" is not pronounced.

On the other hand, there are plenty of examples of the opposite.

In many other text to speech programs the American speaker says "hundred and third"

American saying the "Hundred and First Airborne"
American saying "There is no organisation in the world that can do what the Hundred and First does"
American saying "Hundred and Seventy Third Airborne
American saying "hundred and tenth anniversary", talking about a Harley Davidson
American saying "hundred and tenth anniversary"

So this question is way outside my knowledge of how people around the world do or should pronounce 103rd. From my experience in BrE you can say it:

A hundred and third

or

One hundred and third

and in AmE I have heard examples of both the above with and without the "and".

There may also be more informal vernacular ways of saying it such as "one oh third".