Well first off starting with a tutor in Year 4 suggests that you are organised. I suspect a chat with the tutor will also give you lots of suggestions.
It is very difficult to predict what vocabulary will be required. Some parents go down the route of having their child memorise lists of words, others go for increasing breadth & quality of reading materials or on-line vocabulary quizzes, etc...
I think your idea of reading newspapers is a good one - but possibly shorter articles or digested news (try The Week magazine) - where you can benignly guide them to appropriate subject matter. Other good ideas are science magazines for children - like National Geographic Kids (available most supermarkets) or BBC Wildlife.
To be honest there is another less 'labour intensive' approach - which avoids the Mum is making me study again (long sigh) from 11+ weary child - have them watch documentaries/ nature programmes. Oddly enough my little fish picked up all sorts of vocabulary watching documentaries (Back story is little fish scored 213 - which was enough for KEHG but after a lot of soul search we didn't put this down on our preference list because we opted for our nearby local secondary comprehensive to avoid commuting through the city centre on public transport (something little fish is very frightened of doing on her own), and we were uncertain whether we could rely on the green bus service continuing) . Oddly enough a silly documentary on the origins of the Olympics rather came in handy come the exam. Now that was just blind luck - but the point is watching a wide variety of quality television programmes (Spring/ Autumn Watch/ Sky at Night/ Wonders of the Solar System/ Secrets of Bones/ Fossil Wonderlands/ the splendid Christmas lecture series from the Royal Society etc... can really help with vocabulary in a far less obvious 'Hey, I'm studying for the 11+' way.
As others have suggested read the stickies at the top of the Birmingham/ Walsall/ etc... regional section for advice/ guidance on what you should be doing to prepare. But know that by starting in Year 4 - you're starting far in advance of many parents (certainly I didn't start until summer between Y4/ Y5 - and then only pushing better quality fiction reading).
My advice: Year 4 - focus on core maths skills (so strong calculation skills addition/ subtraction/ multiplication/ division and good concepts of geometry (esp area/ perimeter) and fractions/ percentages/ proportions. Handling Data presented in graphs/ tables and being able to respond to questions about that data is also important.
For English/ VR - Read, read, read. There's a really useful list here: http://www.elevenplusexams.co.uk/advice/english/reading-list
With the work your DC is doing with their tutor as well - you should feel assured that they're making a good start and keep it going (make sure some pennies in the bank of 11+ are being paid in each week) but don't go mad, I really feel that if you make it too much of a slog you'll burn them out. So best approach is a bit each day (30 minutes maybe an hour max) but not overkill. Taking the odd day off - or rewarding them with a treat (going to a movie/ theatre/ concert/ museum/ day out/ etc... also helps.
Our approach was to gradually incorporate workbooks (we went DIY) from start of Year 5, practice papers from after Christmas Y5 were gently introduced and done more seriously after Easter Y5. Remember there's a long summer (between Y5 to Y6) in the run up to the 11+ (however keep an eye out for rumours of the test date moving forward to July of Y5 for 2016, but so far 2015 test date for entry Sept 2016 remains early Sept.). We opted for a short break (one week away) in the summer before Y6 and otherwise were at home working away most mornings. I tried to include breaks (throwing the frisbee or a ball around, etc...) but we put in a solid 2 - 3 hours each day in late July/ August in the run up to the exam. Afternoons were for playing and fresh air - sometimes little fish would be content with a good book, but if she opted to play video games with her sister that was fine too! It was this stage where we discovered what kinds of questions were tricky and would work out strategies to better handle them. I spent afternoons scoring work and trying to focus of final preparations on those areas where questions were consistently wrong.
My main advice is make it about improving ability and not about passing/ failing. You can't guarantee how they will feel (confidence or health) on the day, you can't predict the content of the test (there's always some new wrinkle added each year), how tough the competition from other schools will be or how the new criteria (FSM places) may benefit/ negatively impact on your DC's results.
As I said my little fish doesn't look to be off to KECHG/ KEFW (her preferences) and will be going to an excellent local senior school (we're lucky that we could afford to move into that catchment) - but this process has meant she's read widely, really bedded down her calculation skills through lots of extra practice and has learned that being as prepared as you possibly can be for an exam is a huge benefit (not only in terms of ultimate success, but in terms of confidence going into an exam). It has also meant that her primary is putting her forward for L6 papers - and that she's likely to achieve at least NC L5 at KS2 SATs - which will help place her in top sets at her new senior school (which is also a fantastic outcome from all this hard work). I suspect her new school will have high expectations of little fish and push her to carrying on working hard - which is secretly all I was after.
Hope that helps