I've a connected app in salesforce and I'm using bulk api for downloading xml files but for updates I'm using getupdate rest call, salesforce documentation says that there's a limit of api calls per day. My question is does that limit apply to my connected app or each client? My connected app uses oauth2 for client authentication and api calls
Each Salesforce org has a limit on the total number of API calls per day for that org. It is not specific to your connected app or client.
The number of allowed API calls varies depending on the Salesforce edition and the counts of each type of license, but you can approximate it as 15,000 + (15,000 * the number of Salesforce Platform licenses). The total is capped at 1,000,000 for Enterprise and Professional editions, but uncapped for Unlimited and Performance edition. You can find the full nuance of the calculation at API Request Limits and Allocations.
Note that the limit is not for a calendar day. Like many Salesforce org-wide limits, it's a rolling 24 hour period. It does not reset at midnight. This means that if you exceed the limit, recovery may not be instantaneous depending on the distribution of limits consumption across the 24 hour window.
You should also note that there are a number of limits specific to the Bulk API, including a cap of 10,000 batches per rolling 24 hour period.