Hi,
Have you tried any of the directories for the 1800's in the archives etc?
Here are some directories online,if you haven't already got this link. Whether he's listed in any of them I don't know.
http://www.historicaldirectories.org/I came across the link below with a quick search on google
old pubs, inns and hotels in cardiff and their landlords, there are a few in the 1800's in Bute st, but alas no pubs or hotels with a Josiah Baker in 1871 for Bute St listed on this site below. The list on the below site may come in handy for you though as it names a few of the pubs and hotels in Bute st.
I looked at the 1871 census for 161 bute st at your Baker family, and I can see a landmark of some sorts further on - A London Hotel (looks like it says that) is listed a few doors down from number 161 , it's located at number 169. If you can locate London Hotel today, or yesterday years as it may be long gone by now
perhaps the pub, or hotel before it is your Baker's. London Hotel is listed in the 1855 cardiff directory according to the below link .
It may also be worth you searching on all different terms and ploughing through the results on google to see if you can come up with anything, if you haven't done so already.
I didn't look through all my results as there were far too many of them relating to Bute street, many of no relevance at all. This below link was one of the first results that I found with the terms Bute street and old cardiff pubs - I haven't tried searching on the terms Hotels, inns etc, Bute st.
Have you looked at 161 Bute st in the 1881/1891 and even 1901 ( if the building/hotel still there in 1901 )etc censuses to see if it gives a hotel name at the address of 161 in the later censuses?
Just a few ideas for you, I don't know if you've already tried all this to find it.
Old Cardiff Pubs, Inns and Hotels:
http://hometown.aol.com/PRode18115/pubs.htm Hope this is of some help
P.S.
I'm also coming across a hardware store in my search results at that 161 number today in a Bute Street. I don't know if this hardware shop is the same Hotel in the same Bute street as in the census (?) .Also many house/bulidng numbers change over the years - if it is perhaps the owners of the hardware shop will know what it's name was in the 1800's - in the deeds of the building . I don't know how you would go about doing this.
" 161 Bute Street Treherbert Treorchy Mid Glamorgan CF42 5PE Hardware Retailers in Cardiff"