Will I lose the remaining time for the domain I have with my current registrar if I transfer it to you?

All time remaining on your domain will be transferred - you will not lose it. The domain name expiration date will also be extended for one year (two years for .UK domain names*).
 
However, if your domain expired with your old Registrar, and you renewed it with them, then we suggest you not to transfer it within 45 days of the previous expiry date since it will not add 1 more year to your domain name's expiration date, and you will lose out on the renewal fees paid to the previous Registrar.

*Note: If .UK domain is transferred earlier than 3 months before expiration, the domain name will not be renewed and the money for the transfer will be refunded back to Namecheap account funds.