The Botanical Gardens are north west of the Old Quarter. If you're visiting the Ho Chi Minh mausoleum it's only a short walk away and you can visit the Bac Mon Ancient Citadel on the way. The fee is only 10 cents (2000 Dong) for entrance and another 2000 Dong for parking (if you come by motorbike). The entrance is on Hoang Hoa Tham but If you're coming from the Bac Mon, you can enter from the top corner through the beer garden and have a pint by the first lake if you feel inclined. They'll collect your fee further down the path.
If you come in the morning, you can see many local people who are here for exercise, school children who have class in the park, newly-weds who have their wedding photos taken... In the twilight, there are a lot of couples.
There are some very forlorn-looking monkeys and peacocks in cages dotted around, which is a disappointing touch. However, there are also some really interesting sculptures by Vietnamese artists. Public art spaces not being so common in Hanoi.
The Cache:
When you are in the park you will recognize a small hill with many staircases. This is where you have to search ! Please be careful - the park is not crowded but still there are many muggle. I guess the best thing would be to come with at least one other person and pretend to make some pictures :-).
Small note on cache maintenance: There is a local cache guardian. Name and contact info on file with geocaching.com.