How to pronounce "another thing"

It depends on the accent/dialect. If the dialect is rhotic, it means the /r/ is pronounced in all positions (pre-vocalically, intervocalically and post-vocalically).

However, if the dialect is non-rhotic, it means the /r/ is only pronounced pre-vocalically (before a vowel).

In Standard British English, the /r/ is only pronounced pre-vocalically. In General American English, the /r/ is usually pronounced in all positions.

So the speakers of non-rhotic accent would pronounce 'another thing' without the /r/: /ə'nʌ.ðə.θɪŋ/ while the speakers of rhotic accent might pronounce it with the /r/: /ə'nʌ.ðər.θɪŋ/.

In dictionaries (such as Cambridge online dictionary), 'another' is transcribed as /əˈnʌð.ə^r/, the superscript r means pronounce the r whenever it precedes a word that begins with a vowel.

There are also rhotic accents in Britain, such as the West Country accent and non-rhotic accents in the US such as the New Yowk accent.

When you pronounce "another thing", should you pronounce the "r" at the end of another?

If you want speak with a non-rhotic accent, then don't pronounce it unless it's followed by a vowel.

It depends on which dialect the speakers use. In American English, a final /r/ is indeed pronounced. So in "another thing", the /r/ in pronounced, especially in careful speech (eg. television interview or news reporting). In British English, the /r/ is not pronounced.