As a print out of these two cards is right next to me, here some info.
To answer you question, the card gets activated immediately on the first use, you do not need to wait to the 1st of a month.
I assume you are referring to the aforementioned Unicom prepaid 6 and 12 month card with 3 and 6 GB, respectively?
China Unicom ("liantong") data options
Chinese version found here:
http://3g.10010.com/3gindex/3gfee/wirelesscard.html
Name:
Prepaid, 3 GB, 6 months, 50 text msgs: 预付费3GB半年卡
Ask specifically for this card in a Unicom store.
Terms:
- 无线上网卡3GB半年卡套餐包含的流量和赠送的短信在180天内有效，6GB年卡套餐包含的流量和赠送的短 信在360天内有效。
- 在此期间内，如套餐内包含的流量用完，则有效期提前结束，未使用完的赠送短信作废，不能再使用。超过期限后 ，套餐未使用完的流量和未使用完的赠送短信同时作废，不能再使用。
- 本产品只允许使用激活，不允许充值激活。当用户首次上网或首次发送短信时激活本产品，其它方式 均不能激活。
- 本产品使用激活后立即扣除套餐费用并赠送相应的短信条数。
- 本产品的默认有效期为产品激活次日零时起计算的累计天数。有效期到期后，用户充值后可以继续使 用。
In short: The period starts running the first time you activate the card. You can only activate it by sending a text or using the internet. You cannot top up the card during the period. Any remaining credit after 180 or 360 days becomes invalid.
All prepaid options require a passport, nothing more.