I have just been prescribed montelukast. Will this help with inflammation? Rheumatoid arthritis?

I know you posted this question almost a decade ago, but currently, in 2019, researchers are testing that very question. It was a brilliant question you asked and it was literally ahead of its time.

Anyhow, here is a resource stating the science behind the theory and why it might work:
https://www.ncbi.nlm.nih.gov/pubmed/29890415

And here is an announcement that phase 1 of trials on the effectiveness of Montelukast, combined with some other drugs, in reducing the inflammation of rherumatoid arthritis have been completed:
https://www.drugbank.ca/drugs/DB00471/clinical_trials?conditions=DBCOND0027961&phase=1&purpose=basic_science&status=completed

Here are the criteria for the study:
https://clinicaltrials.gov/ct2/show/NCT03131973

So far, I've found no results or drug trial reports, but there could definitely be something to montelukast positively affecting Rheumatoid Arthritis symptoms.

No Montelukast won't help with inflammation due to rheumatoid arthritis.

Montelukast is a leukotriene inhibitor. Leukotrienes are chemicals your body releases when you breathe in an allergen (such as pollen). These chemicals cause swelling in your lungs and tightening of the muscles around your airways, which can result in asthma symptoms.

It binds to cysteinyl leukotriene (CysLT 1 ) receptor in the upper and lower airways to prevent leukotriene-mediated effects associated with asthma and allergic rhinitis. So it very specifically works in the lungs.

Montelukast is used to prevent asthma attacks in adults and children as young as 12 months old. It is also used to relieve runny nose and sneezing caused by allergies in adults and children as young as 6 months old.

https://www.drugs.com/mtm/montelukast.html

https://www.drugs.com/ppa/montelukast.html