TL;DR Tolkien had various versions and lengths he played with. Deciding between 3000 and 5000 Valian Years for the Ages of the Valar, deciding between 9.58, 10 or 144 Years of the Sun for each Year of the Trees. This gives us a range from 43000 years old to 733000 years old to millions of years.
The main thing missing in @apollo's answer is the fact she's quoted one continuum of Tolkien's Legendarium, that in which Tolkien uses slightly over 9 for the number of Years of the Sun in one Year of the Trees (approx. 9.58). Tolkien wrote many versions of the stories that conflict and contradict each other. This has created various continuums in which his tales lay intertwined, similar but not the same.
In the below quotes, AAm is the Annals of Aman, AV2 is the Annals of Valinor Part 2 (written pre-LotR)
As this sentence was first written in the draft text for the beginning of AAm (the rewriting of A V 2) it read: 'each such year is in length even as are ten years of the Sun that is now'; i.e., my father still retained the old much simpler computation going back through AV 2 (V.110) to AV 1 (IV.263). This was changed on the draft text to 'each such year is longer than are nine years of the Sun that is now'.
From the above we can see that in the earlier legendarium, Tolkien had his mind set on 10 Years of the Sun (Y.S.) to one Year of the Trees (Y.T.). This only slightly extends the "age" of Middle-earth/Arda. That puts us the coming of the Valar about 3000 years earlier than in apollo's answer. So we see in this older version of the Legendarium Tolkien had intended it to be a slightly older timeline. Extending the length before the suns rise by about 3000 years (or 300 Valian Years)
Below we have further evidence of Tolkien using the 10 Y.S. = 1 Y.T.
five and thirty Years of the Valar (which is like unto three hundred and thirty-five of our years)
For well-nigh one hundred of the years of our time (though that be but ten of the Years of the Valar) they dwelt in Tol Eressea.
And even as they came the First Ages of the World were ended;(24) and these are reckoned as 30000 years or 3000 years of the Valar whereof the First Thousand was before the Trees, and Two Thousand save nine were Years of the Trees or of the Holy Light, which lived after and lives yet only in the Silmarils. And the Nine are the Years of Darkness or the Darkening of Valinor.
The above is interesting to note, as although it gives us further confirmation of the 10 = 1 rule, it also informs us of the fact in another version of the Legendarium the Ages of the Valar, before the rising of the sun only lasted 3000 years. As opposed to the version mentioned by apollo which lasted 5000 years. This would make the age of Arda around 43000 years old. A whole 17000 years younger than that which was proposed in Christopher's edits.
It was indeed at the landing of Fëanor three hundred and sixty-five long years of the Valar(1) since the Noldor had passed over the Sea and left the Teleri behind them. Now that time was in length well nigh as three thousand and five hundred years of the Sun.
The War of the Jewels
In the War of the Jewels we get an example of the ~9.58 Y.S. = 1 Y.T. This seems to be the most accepted answer, as quoted on tolkiengateway, as well as elsewhere. Providing us with the ~60,000 year old Arda as seen above.
“In notes not given in this book, in which my father was calculating on this basis the time of the Awakening of Men, he expressly stated that 144 Sun Years = 1 Valian Year expressly stated that 144 Sun Years = 1 Valian Year (in this connection see Appendix D to The Lord of the Rings: 'It seems clear that the Eldar in Middle-earth ... reckoned in long periods, and the Quenya word yen... really means 144 of our years').
In other notes, Tolkien thought of 1 Y.T. to be equal to 144 Y.S. (likely due to the Elves' obsession with sixes and twelves). This would put the age of Arda at ~733000 years. With the Ages of the Valar taking up around 720000 of those years.
These longer years seems to be something Tolkien played with later in his life. Although I couldn't find a source for it, the Tolkien scholar, Michael Martinez, speaks of notes where Tolkien considered making Arda millions or billions of years old, which would allow for the reshaping of the world to fit what we have today.