The cost of living in Leiden (and the Netherlands in general) is not cheap, but it’s usually manageable on an average student budget. On areas like public transport you can save a good deal of money compared to larger cities, since you’ll probably just cycle most of the time! Another bonus is that since Leiden is a very student-heavy city, there are often great drinks and food promotions across the city to appeal to the student budget.
Accommodation is always the big hitter. Rent could be anywhere from €350-600 a month depending on the location, size and furnishings. In addition you’ll have to pay utility bills (like heating and electricity) and internet, which could be around €100 per month for the apartment. Remember that if you get an unfurnished place, you’ll save money on rent but will also have to buy expensive items like beds, sofas and all your kitchen equipment - you may end up spending more in the long run.
Groceries aren’t too expensive here. The Saturday food market in Leiden is one of the best in the country, and a superb way to get loads of your fruit, veg and other bits and pieces. There are also supermarkets where you’ll get tinned foods, probably a lot of your meat, pasta and so on - these are fairly priced, but of course how much you spend is really up to how exciting your diet is. €200 a month should see you eating well if you make most of your own meals.
Since you’ll cycle almost everywhere, a bike is probably your most important public transport investment. A second hand bike will go for between €50-100 if you want something good but not over the top. Along with a strong lock and maybe a puncture repair kit, this will be the bulk of your local transport costs in a one-er.
For getting local buses or public transport in other Dutch cities, you’ll need to buy an OV-Chipkaart. This is the Dutch travel card and is valid on everything: buses, trams, metro and trains. You need a €20 minimum balance on the card and each journey is charged based on the exact time and distanced travelled, rather than a flat "per journey" fee.