I think the following are the correct answers to your questions (please bare in mind this is just my opinion)
1) you would say 'She won first place' or 'She came first in the exam'. You wouldn't say 'She won 'the' first place'
2) I would write 1/3 or 1:3 if you are referring to mathematical terms. it would not be incorrect to write 'one third'
3) I would write 800m race. This is as long as everyone knows the 'm' signifies metres. If you want to write the full version it would be '800 metre race' (without the 's'). Please note American way of spelling is meter, UK way is metre.
Hope this helps
Student or Learner